Rivalry2

HTML Diff

1 added 2 removed

Original 2026-01-01

Modified 2026-02-28

1 - <p>145 Learners</p>

1 + <p>180 Learners</p>

2 <p>Last updated on<strong>October 30, 2025</strong></p>

3 <p>Zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation f(x) = 0. In this article, we will learn about the zeros of a quadratic polynomial and the methods to find them, such as factorization and the quadratic formula.</p>

4 <h2>What are Zeros of Quadratic Polynomials?</h2>

5 <p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>

6 <p>▶</p>

7 <p>The value of x that makes a<a>quadratic polynomial</a>equal to zero is known as the zero of a quadratic polynomial.</p>

8 <p>For a quadratic polynomial of the form f(x) = ax2 + bx + c, where a ≠ 0, the zeros of the quadratic polynomial are the values x for which f(x) = 0.</p>

9 <p>The zeros of a quadratic polynomial are also known as roots or solutions of the polynomial. </p>

10 <h2>What is a Quadratic Polynomial?</h2>

11 <h2>How to find Zeros of Quadratic Polynomials?</h2>

12 <p>A quadratic<a>polynomial</a>can have a maximum of two zeros because its highest degree is 2. Now, let’s learn how to find the zeros of a quadratic polynomial.</p>

13 <p>The common methods used to find the zeros of a quadratic polynomial are: </p>

14 <ol><li>Factorization Method</li>

15 <li>Quadratic Formula</li>

16 </ol><ul><li><strong>Factorization Method</strong>In the factorization method, to find the zeros of a quadratic polynomial, we<a>set</a>the<a>equation</a>as the<a>product</a>of two linear polynomials by factoring them.</li>

17 </ul><p>Let's practice. </p>

18 <p>Find the zeros of a quadratic polynomial x2 - 7x + 12 = 0</p>

19 <ol><li><strong>Step 1: </strong>Arrange the<a>polynomial in</a><a>standard form</a>(ax2 + bx + c = 0) Standard form: x2 - 7x + 12 = 0 Here, a = 1, b = -7, and c = 12 </li>

20 <li><strong>Step 2:</strong>Finding two<a></a><a>numbers</a>Now find two numbers whose product is ac and<a>sum</a>is b Here, the numbers are -3 and -4, as -3 × -4 = 12 and -3 + -4 = -7 </li>

21 <li><strong>Step 3:</strong>Splitting the middle term Here, we split 7x as 3x and 4x, then the quadratic polynomial becomes: x -2 - 3x - 4x + 12 = 0 </li>

22 <li><strong>Step 4</strong>: Grouping and factoring Now we factor, x 2 - 3x - 4x + 12 = 0 as: \((x^2 - 3x) - (4x - 12) = 0\\ x(x - 3) - 4(x - 3) = 0\\ (x - 3)(x - 4) = 0\) </li>

23 <li><strong>Step 5:</strong>Solve the equation To find the value of x, we solve the equation: \(x - 3 = 0 → x = 3\\ x - 4 = 0 → x = 4\)</li>

24 </ol><ul><li><strong>Quadratic Formula</strong>The quadratic formula to find the value of zeros of a quadratic polynomial in the form: ax 2 + bx + c = 0 is:</li>

25 </ul><p>Let's practice. </p>

26 <p>Find the zeros of a quadratic polynomial 3x2 - 5x + 2 = 0 </p>

27 <ol><li>Here, a = 3 b = -5 c = 2 </li>

28 <li>Substituting the values in the formula: \(x = \frac{-b \ ± \ \sqrt{b^2 - 4ac}}{2a}\)<p>\(x = \frac{- (-5) \ ± \ \sqrt{(-5)^2 \ -\ 4 × 3 × 2}}{2 × 3}\\ x = \frac{5 \ ± \ \sqrt{25 - 24}}{6}\\ x = {5 ± 1\over 6}\\ x = {5+1\over 6} \ and \ x = {5-1\over 6}\\ x = {6\over 6} = 1\\ x = {4\over 6} = ⅔\)</p>

29 </li>

30 </ol><p>The zeros of the quadratic polynomial are 1 and 2/3 </p>

31 <h3>Explore Our Programs</h3>

32 - <p>No Courses Available</p>

33 <h2>Nature of Zeros of Quadratic Polynomial</h2>

32 <h2>Nature of Zeros of Quadratic Polynomial</h2>

34 <p>The nature of zeros of a quadratic<a>polynomial</a>refers to the nature of zeros. It tells whether the zeros are real, distinct, or<a></a><a>complex numbers</a>.</p>

33 <p>The nature of zeros of a quadratic<a>polynomial</a>refers to the nature of zeros. It tells whether the zeros are real, distinct, or<a></a><a>complex numbers</a>.</p>

35 <p>The nature of zeros is determined using the<a>discriminant</a>(D) of the quadratic polynomial: D = b 2 - 4ac. </p>

34 <p>The nature of zeros is determined using the<a>discriminant</a>(D) of the quadratic polynomial: D = b 2 - 4ac. </p>

36 <p>The following table shows the nature of zeros according to the value of discriminant.</p>

35 <p>The following table shows the nature of zeros according to the value of discriminant.</p>

37 <p><strong>Value of discriminant </strong></p>

36 <p><strong>Value of discriminant </strong></p>

38 <p><strong>Nature of zeros</strong></p>

37 <p><strong>Nature of zeros</strong></p>

39 <p>If D < 0</p>

38 <p>If D < 0</p>

40 <p>The zeros are complex</p>

39 <p>The zeros are complex</p>

41 <p>If D > 0 </p>

40 <p>If D > 0 </p>

42 <p>The zeros are<a>real</a>and distinct </p>

41 <p>The zeros are<a>real</a>and distinct </p>

43 <p>If D = 0</p>

42 <p>If D = 0</p>

44 <p>The zeros are real and equal </p>

43 <p>The zeros are real and equal </p>

45 <h2>Sum and product of Zeros of Quadratic Polynomial</h2>

44 <h2>Sum and product of Zeros of Quadratic Polynomial</h2>

46 <p>The sum and product of the zeros of a quadratic polynomial show the relationship between its coefficients and<a></a><a>roots of quadratic equation</a>.</p>

45 <p>The sum and product of the zeros of a quadratic polynomial show the relationship between its coefficients and<a></a><a>roots of quadratic equation</a>.</p>

47 <p>Let α and β be the zeros of the<a>quadratic equation</a> ax2 + bx + c = 0. </p>

46 <p>Let α and β be the zeros of the<a>quadratic equation</a> ax2 + bx + c = 0. </p>

48 <ul><li>The sum of zeros: α + β = -b/a </li>

47 <ul><li>The sum of zeros: α + β = -b/a </li>

49 <li>The product of zeros: αβ = c/a </li>

48 <li>The product of zeros: αβ = c/a </li>

50 </ul><h2>Graphical Representation of Zeros of Quadratic Polynomial</h2>

49 </ul><h2>Graphical Representation of Zeros of Quadratic Polynomial</h2>

51 <p>Let’s now learn how to find the zeros of a<a>quadratic polynomial</a>using a graph.</p>

50 <p>Let’s now learn how to find the zeros of a<a>quadratic polynomial</a>using a graph.</p>

52 <ul><li>The graph of a quadratic polynomial forms a parabola. </li>

51 <ul><li>The graph of a quadratic polynomial forms a parabola. </li>

53 <li>The x-intercepts of this graph are the points where the graph intersects the x-axis. </li>

52 <li>The x-intercepts of this graph are the points where the graph intersects the x-axis. </li>

54 <li>The point of intersection represents the real zeros of a quadratic polynomial. </li>

53 <li>The point of intersection represents the real zeros of a quadratic polynomial. </li>

55 </ul><h2>Tips and Tricks to Master Zeros of Quadratic Polynomial</h2>

54 </ul><h2>Tips and Tricks to Master Zeros of Quadratic Polynomial</h2>

56 <p>Zeroes of quadratic polynomials can be difficult for students of smaller grades. So, here are some essential tips and tricks to make it easy:</p>

55 <p>Zeroes of quadratic polynomials can be difficult for students of smaller grades. So, here are some essential tips and tricks to make it easy:</p>

57 <ol><li>To recognize if the given equation is a quadratic, check its<a>degree</a>. </li>

56 <ol><li>To recognize if the given equation is a quadratic, check its<a>degree</a>. </li>

58 <li>When factorizing, ensure that the signs in the equation do not change. </li>

57 <li>When factorizing, ensure that the signs in the equation do not change. </li>

59 <li>If c = 0, then the zeroes of equation ax² + bx is x = 0 and \(x = { -b\over a}\) </li>

58 <li>If c = 0, then the zeroes of equation ax² + bx is x = 0 and \(x = { -b\over a}\) </li>

60 <li>When finding<a>factors</a>seems difficult, directly use the<a>formula</a> </li>

59 <li>When finding<a>factors</a>seems difficult, directly use the<a>formula</a> </li>

61 <li>Memorize the formula of<a>discriminant</a>, \(D = b^2 - 4ac\), to find the<a></a><a>nature of roots</a>or zeroes.</li>

60 <li>Memorize the formula of<a>discriminant</a>, \(D = b^2 - 4ac\), to find the<a></a><a>nature of roots</a>or zeroes.</li>

62 </ol><p><strong>Parent Tips:</strong></p>

61 </ol><p><strong>Parent Tips:</strong></p>

63 <ul><li>Help your child memorize all formulas.</li>

62 <ul><li>Help your child memorize all formulas.</li>

64 <li>You can check your child’s answer manually using the sum and product of the zeros’ formula. </li>

63 <li>You can check your child’s answer manually using the sum and product of the zeros’ formula. </li>

65 <li>You can also use<a></a><a>polynomial equation</a>solver calculator to verify the zeros. </li>

64 <li>You can also use<a></a><a>polynomial equation</a>solver calculator to verify the zeros. </li>

66 </ul><h2>Real-World Applications of Zeros of a Quadratic Polynomial</h2>

65 </ul><h2>Real-World Applications of Zeros of a Quadratic Polynomial</h2>

67 <p>The zeros of a quadratic polynomial are used in fields like physics, engineering, architecture, finance, etc. We will explore how the quadratic formula is applied in real-life applications.</p>

66 <p>The zeros of a quadratic polynomial are used in fields like physics, engineering, architecture, finance, etc. We will explore how the quadratic formula is applied in real-life applications.</p>

68 <ol><li>In physics, the zeros of a quadratic polynomial are used to understand the projectile motion. For example, to find the height of a ball if it is thrown upwards, we use the quadratic polynomial. </li>

67 <ol><li>In physics, the zeros of a quadratic polynomial are used to understand the projectile motion. For example, to find the height of a ball if it is thrown upwards, we use the quadratic polynomial. </li>

69 <li>In structural design, a quadratic polynomial is used to model the parabolic arches or support structures. The zeros help us model a structure, like a beam or bridge, where there is no movement or where the stress changes. </li>

68 <li>In structural design, a quadratic polynomial is used to model the parabolic arches or support structures. The zeros help us model a structure, like a beam or bridge, where there is no movement or where the stress changes. </li>

70 <li>In finance, the zeros of a quadratic polynomial are used to find the time of repayment of a loan or to calculate the number of payments needed. </li>

69 <li>In finance, the zeros of a quadratic polynomial are used to find the time of repayment of a loan or to calculate the number of payments needed. </li>

71 <li>To describe the population growth or decline, we use the quadratic polynomial. The zeros of a quadratic polynomial help identify when the population might start or stop growing, or when it drops to a critical low point. </li>

70 <li>To describe the population growth or decline, we use the quadratic polynomial. The zeros of a quadratic polynomial help identify when the population might start or stop growing, or when it drops to a critical low point. </li>

72 <li>In video games, animations, VR simulations, and 3D modeling, designers and programmers use<a>quadratic equations</a>to model jumping, falling, throwing or projectile movements. Zeroes determine when an object hits the ground or returns to the original point.</li>

71 <li>In video games, animations, VR simulations, and 3D modeling, designers and programmers use<a>quadratic equations</a>to model jumping, falling, throwing or projectile movements. Zeroes determine when an object hits the ground or returns to the original point.</li>

73 </ol><h2>Common Mistakes and How to Avoid Them in Zeros of Quadratic Polynomial</h2>

72 </ol><h2>Common Mistakes and How to Avoid Them in Zeros of Quadratic Polynomial</h2>

74 <p>When finding the zeros of a quadratic polynomial, students make errors. Here are some common mistakes students make when finding the zeros of a quadratic polynomial, and the tips to avoid them. </p>

73 <p>When finding the zeros of a quadratic polynomial, students make errors. Here are some common mistakes students make when finding the zeros of a quadratic polynomial, and the tips to avoid them. </p>

75 <h3>Problem 1</h3>

74 <h3>Problem 1</h3>

76 <p>Find the zeros of a quadratic polynomial: x2 -5x + 6</p>

75 <p>Find the zeros of a quadratic polynomial: x2 -5x + 6</p>

77 <p>Okay, lets begin</p>

76 <p>Okay, lets begin</p>

78 <p>Here, x = 2 and x = 3 </p>

77 <p>Here, x = 2 and x = 3 </p>

79 <h3>Explanation</h3>

78 <h3>Explanation</h3>

80 <ol><li>To find the zeros of the polynomial x2 - 5x + 6, we factor the polynomial x2 - 5x + 6 = (x - 2)(x - 3) </li>

79 <ol><li>To find the zeros of the polynomial x2 - 5x + 6, we factor the polynomial x2 - 5x + 6 = (x - 2)(x - 3) </li>

81 <li>The factors used are -2 and -3 because their product is 6 (which equals ac) and their sum is -5 (which equals b) </li>

80 <li>The factors used are -2 and -3 because their product is 6 (which equals ac) and their sum is -5 (which equals b) </li>

82 <li>Solving the equations: x - 2 = 0 → x = 2 x - 3 = 0 → x = 3 </li>

81 <li>Solving the equations: x - 2 = 0 → x = 2 x - 3 = 0 → x = 3 </li>

83 </ol><p>Well explained 👍</p>

82 </ol><p>Well explained 👍</p>

84 <h3>Problem 2</h3>

83 <h3>Problem 2</h3>

85 <p>Find the zeros of x2 - 4x + 4</p>

84 <p>Find the zeros of x2 - 4x + 4</p>

86 <p>Okay, lets begin</p>

85 <p>Okay, lets begin</p>

87 <p>The polynomial has one real zero, x = 2. </p>

86 <p>The polynomial has one real zero, x = 2. </p>

88 <h3>Explanation</h3>

87 <h3>Explanation</h3>

89 <ol><li>To find the root of the polynomial x2 - 4x + 4, we use the factorization method. x2 - 4x + 4 = (x - 2)2 </li>

88 <ol><li>To find the root of the polynomial x2 - 4x + 4, we use the factorization method. x2 - 4x + 4 = (x - 2)2 </li>

90 <li>So, x - 2 = 0 x = 2</li>

89 <li>So, x - 2 = 0 x = 2</li>

91 </ol><p>Well explained 👍</p>

90 </ol><p>Well explained 👍</p>

92 <h3>Problem 3</h3>

91 <h3>Problem 3</h3>

93 <p>Find the sum and product of the quadratic polynomial: x2 + 2x + 5</p>

92 <p>Find the sum and product of the quadratic polynomial: x2 + 2x + 5</p>

94 <p>Okay, lets begin</p>

93 <p>Okay, lets begin</p>

95 <p> The sum of the quadratic polynomial is -2, and the product of the quadratic polynomial is 5 </p>

94 <p> The sum of the quadratic polynomial is -2, and the product of the quadratic polynomial is 5 </p>

96 <h3>Explanation</h3>

95 <h3>Explanation</h3>

97 <ol><li>Here, a = 1, b = 2, and c = 5 </li>

96 <ol><li>Here, a = 1, b = 2, and c = 5 </li>

98 <li>To find the sum of its zeros, we use the formula: -b/a -b/a = -2/1 = -2 </li>

97 <li>To find the sum of its zeros, we use the formula: -b/a -b/a = -2/1 = -2 </li>

99 <li>To find the product of the zeroes, we use the formula: c/a c/a = 5/1 = 5</li>

98 <li>To find the product of the zeroes, we use the formula: c/a c/a = 5/1 = 5</li>

100 </ol><p>Well explained 👍</p>

99 </ol><p>Well explained 👍</p>

101 <h3>Problem 4</h3>

100 <h3>Problem 4</h3>

102 <p>Find the zeros of x2 + x - 6</p>

101 <p>Find the zeros of x2 + x - 6</p>

103 <p>Okay, lets begin</p>

102 <p>Okay, lets begin</p>

104 <p> x = 2 and x = -3 </p>

103 <p> x = 2 and x = -3 </p>

105 <h3>Explanation</h3>

104 <h3>Explanation</h3>

106 <p>To find the zeros of a quadratic polynomial, we use the formula: \(x = \frac{-b \ ± \ \sqrt{b^2 - 4ac}}{2a}\) Here, a = 1, b = 1 c = -6</p>

105 <p>To find the zeros of a quadratic polynomial, we use the formula: \(x = \frac{-b \ ± \ \sqrt{b^2 - 4ac}}{2a}\) Here, a = 1, b = 1 c = -6</p>

107 <p>\(x = \frac{-1 \ ± \ \sqrt {1^2 \ - \ 4(1)(-6)}}{2a}\) \(x = {-1 \ ± \sqrt{1\ + 24} \over 2}\) \(x = {-1 \ ± \ \sqrt{25} \over 2}\)</p>

106 <p>\(x = \frac{-1 \ ± \ \sqrt {1^2 \ - \ 4(1)(-6)}}{2a}\) \(x = {-1 \ ± \sqrt{1\ + 24} \over 2}\) \(x = {-1 \ ± \ \sqrt{25} \over 2}\)</p>

108 <p>So, \(x = {-1 \ + \ 5 \over 2}\) and\( x = {-1 \ - \ 5 \over 2}\)</p>

107 <p>So, \(x = {-1 \ + \ 5 \over 2}\) and\( x = {-1 \ - \ 5 \over 2}\)</p>

109 <p>x = 4/2 = 2 x = -6/2 = -3</p>

108 <p>x = 4/2 = 2 x = -6/2 = -3</p>

110 <p>Well explained 👍</p>

109 <p>Well explained 👍</p>

111 <h3>Problem 5</h3>

110 <h3>Problem 5</h3>

112 <p>Find the zeros of x2 + 6x + 9 and verify the relationship between zeros and coefficients of the polynomial.</p>

111 <p>Find the zeros of x2 + 6x + 9 and verify the relationship between zeros and coefficients of the polynomial.</p>

113 <p>Okay, lets begin</p>

112 <p>Okay, lets begin</p>

114 <p> x = -3 and x = -3. The relationship between zeros and coefficients holds. </p>

113 <p> x = -3 and x = -3. The relationship between zeros and coefficients holds. </p>

115 <h3>Explanation</h3>

114 <h3>Explanation</h3>

116 <p>To find the zeros of a quadratic polynomial, we use the formulas: \(x = \frac{-b \ ± \ \sqrt{b^2 - 4ac}}{2a}\) Where, a = 1, b = 6, c = 9</p>

115 <p>To find the zeros of a quadratic polynomial, we use the formulas: \(x = \frac{-b \ ± \ \sqrt{b^2 - 4ac}}{2a}\) Where, a = 1, b = 6, c = 9</p>

117 <p>\(x = \frac{-6 \ ± \ \sqrt{6^2 - 4(1)(9)}}{2(1)}\) \(x = \frac{-6 \ ± \ \sqrt{36 - 36}}{2(1)}\) \(x = \frac{-6 \ ± \ 0}{2(1)}\) x = -6/2 = -3</p>

116 <p>\(x = \frac{-6 \ ± \ \sqrt{6^2 - 4(1)(9)}}{2(1)}\) \(x = \frac{-6 \ ± \ \sqrt{36 - 36}}{2(1)}\) \(x = \frac{-6 \ ± \ 0}{2(1)}\) x = -6/2 = -3</p>

118 <p>So, the value of zeros are x = -3 and x = -3</p>

117 <p>So, the value of zeros are x = -3 and x = -3</p>

119 <ul><li>The sum of zeros = -3 + -3 = -6</li>

118 <ul><li>The sum of zeros = -3 + -3 = -6</li>

120 <li>The product of zeros = -3 × -3 = 9</li>

119 <li>The product of zeros = -3 × -3 = 9</li>

121 </ul><p>To verify the relationship between zeros and coefficients, for a quadratic polynomial ax2 + bx + c: </p>

120 </ul><p>To verify the relationship between zeros and coefficients, for a quadratic polynomial ax2 + bx + c: </p>

122 <ul><li>Sum of zeros = -b/a = -6/1 = -6</li>

121 <ul><li>Sum of zeros = -b/a = -6/1 = -6</li>

123 <li>Product of zeros = c/a = 9/1 = 9</li>

122 <li>Product of zeros = c/a = 9/1 = 9</li>

124 </ul><p>So, the relationship between zeros and coefficients of the polynomial is verified. </p>

123 </ul><p>So, the relationship between zeros and coefficients of the polynomial is verified. </p>

125 <p>Well explained 👍</p>

124 <p>Well explained 👍</p>

126 <h2>FAQs on Zeros of Quadratic Polynomial</h2>

125 <h2>FAQs on Zeros of Quadratic Polynomial</h2>

127 <h3>1.What is the way my child can verify if the calculated value of zeroes are correct?</h3>

126 <h3>1.What is the way my child can verify if the calculated value of zeroes are correct?</h3>

128 <p>To check if the zeroes are correct, children can use the formula of<a>sum and product of zeroes</a>. </p>

127 <p>To check if the zeroes are correct, children can use the formula of<a>sum and product of zeroes</a>. </p>

129 <ul><li>Sum of zeroes = \(-\frac{b}{a} \)</li>

128 <ul><li>Sum of zeroes = \(-\frac{b}{a} \)</li>

130 <li>Product of zeroes = \(\frac{c}{a}\)</li>

129 <li>Product of zeroes = \(\frac{c}{a}\)</li>

131 </ul><h3>2.Why is it necessary for my child to learn about zeroes?</h3>

130 </ul><h3>2.Why is it necessary for my child to learn about zeroes?</h3>

132 <p>Because it will help your child to:</p>

131 <p>Because it will help your child to:</p>

133 <ul><li>Connect<a></a><a>algebra</a>and<a></a><a>geometry</a>(zeroes correspond to where the parabola cuts the x-axis)</li>

132 <ul><li>Connect<a></a><a>algebra</a>and<a></a><a>geometry</a>(zeroes correspond to where the parabola cuts the x-axis)</li>

134 <li>Solve real-life problems like finding time, distance, or<a>profit</a>when something “reaches zero”</li>

133 <li>Solve real-life problems like finding time, distance, or<a>profit</a>when something “reaches zero”</li>

135 <li>Build a strong foundation for advanced-level algebra,<a></a><a>calculus</a>, and physics </li>

134 <li>Build a strong foundation for advanced-level algebra,<a></a><a>calculus</a>, and physics </li>

136 </ul><h3>3.How can parents help children visualize zeroes?</h3>

135 </ul><h3>3.How can parents help children visualize zeroes?</h3>

137 <p>With the help of a graph, show how the parabola touches the x-axis, and the points are the zeroes.</p>

136 <p>With the help of a graph, show how the parabola touches the x-axis, and the points are the zeroes.</p>

138 <p>You can also use examples like: “At what time will the height of a ball be zero again after being thrown?” - that’s finding a zero! </p>

137 <p>You can also use examples like: “At what time will the height of a ball be zero again after being thrown?” - that’s finding a zero! </p>

139 <h3>4.What common mistakes can my child make?</h3>

138 <h3>4.What common mistakes can my child make?</h3>

140 <p>Here are a few mistakes</p>

139 <p>Here are a few mistakes</p>

141 <ol><li>Misinterpreting zeroes as coefficients or constants</li>

140 <ol><li>Misinterpreting zeroes as coefficients or constants</li>

142 <li>Incorrectly using formula</li>

141 <li>Incorrectly using formula</li>

143 <li>Not considering the cases when some<a>quadratics</a>have no real zeroes</li>

142 <li>Not considering the cases when some<a>quadratics</a>have no real zeroes</li>

144 <li>Not verifying their answers. </li>

143 <li>Not verifying their answers. </li>

145 </ol><h3>5.How can I relate this to real life?</h3>

144 </ol><h3>5.How can I relate this to real life?</h3>

146 <p>You can use examples such as:</p>

145 <p>You can use examples such as:</p>

147 <ul><li>When a ball thrown from a height hits the ground, and the height becomes zero.</li>

146 <ul><li>When a ball thrown from a height hits the ground, and the height becomes zero.</li>

148 <li>When profit = 0, the company breaks even.</li>

147 <li>When profit = 0, the company breaks even.</li>

149 <li>When the distance left to cover equals zero, the journey ends.</li>

148 <li>When the distance left to cover equals zero, the journey ends.</li>

150 <li>Each of these points denotes a zero, which is a point where something reaches nothing! </li>

149 <li>Each of these points denotes a zero, which is a point where something reaches nothing! </li>

151 </ul>

150 </ul>