Lesson 8 focuses on the sum and product of roots in quadratic equations, providing essential formulas for students studying algebra. It covers the quadratic equation format ax² + bx + c = 0 and explains how to calculate the sum and product of the roots. This lesson includes examples and problem sets to reinforce understanding, making it ideal for high school students preparing for exams. Key concepts include the derivation of the quadratic equation from its roots and practical applications in solving quadratic equations.

Key Points

  • Explains the formulas for sum and product of roots in quadratic equations.
  • Includes examples of quadratic equations such as x² + 5x + 4 = 0.
  • Provides problem sets for practice on calculating roots.
  • Covers the derivation of quadratic equations from given roots.
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Quadratics
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Quadratics
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12
b
r +r = -
a
5
=-
1
12
12
r+
rr
r
+
= -5
12
c
r r =
a
4
=
1
12
12
r + r
r r = 4
12
b
r +r = -
a
( 8)
=-
1
2
2
1
1
r
rr
+r
+
=8
12
c
r r =
a
15
=
1
12
12
r
r+
r
r
= -15
Given the quadratic equation ax
2
+ bx + c = 0, the sum and product of the roots r
1
and
r
2
can be obtained by:
Sum of the Roots Product of the Roots
12
b
r + r = -
a
12
c
r r =
a
The quadratic equation with roots r
1
and r
2
can be obtained by:
x
2
– (r
1
+ r
2
)x + (r
1
● r
2
) = 0
(a) x
2
+ 5x + 4 = 0 a = 1; b = 5; c = 4
(b) x
2
– 8x – 15 = 0 a = 1; b = -8; c = -15
LESSON 8:
Sum and Product of Roots of
Quadratic Equations
MODULE 1: QUADRATIC EQUATIONS & INEQUALITIES
Example 1:
equations.
(a) x
2
+ 5x + 4 = 0 (c) 3x
2
– 2x – 1 = 0
Solution:
12
b
r +r = -
a
( 2)
=-
3
2
2
1
1
2
r
r
+r
+
=
r
3
12
c
r r =
a
1
=
3
2
2
1
1
1
r
r
r
r
=-
+
3
(c) 3x
2
– 2x – 1 = 0 a = 3; b = -2; c = -1
We have to solve for the sum and product first.
(a) 3 and -5
r
1
+ r
2
= 3 + (-5) r
1
● r
2
= (3)(-5)
= -2 = -15
Substituting we have,
x
2
– (r
1
+ r
2
)x + (r
1
● r
2
) = 0
x
2
+ 2x – 15 = 0
Therefore, the quadratic equation whose roots are 3 and -5 is x
2
+ 2x – 15 = 0.
(b)
1
3
and
2
5
r
1
+ r
2
=
12
35
5(1) -3(2)
15
r
1
● r
2
=
12
35
=
1
15
=
2
15
x
2
– (r
1
+ r
2
)x + (r
1
● r
2
) = 0
2
12
x x 0
15 15
15x
2
+ x – 2 = 0
Therefore, the quadratic equation whose roots are
1
3
and
2
5
is 15x
2
+ x – 2 = 0.
Example 2:
(a) 3 and -5 (c) and
(b) and
(d) and
Solution:
(c)
3
and
3
r
1
+ r
2
=
3
3
r
1
● r
2
=
33
= 0 = -3
x
2
– (r
1
+ r
2
)x + (r
1
● r
2
) = 0
x
2
– 3 = 0
The quadratic equation whose roots are
3
and
3
is x
2
– 3 = 0.
(d)
22
and
22
r
1
+ r
2
=
22
+
22
r
1
● r
2
=
2 2 2 2
= 4 = 4 – 2
= 2
x
2
– (r
1
+ r
2
)x + (r
1
● r
2
) = 0
x
2
– 4x + 2 = 0
The quadratic equation whose roots are
22
and
22
is x
2
– 4x + 2 = 0.
A. Determine the sum and product of roots of the following quadratic equations.
ax
2
+ bx + c = 0
Sum of the Roots
Product of the Roots
1. n
2
– 5n + 6 = 0
2. y
2
– 7y – 2 = 0
3. a
2
+ 4a – 5 = 0
4. 2x
2
+ x – 5 = 0
5. 3x
2
– x – 3 = 0
6. 2p
2
– 2p – 1 = 0
7. 3x
2
– 2x – 8 = 0
Problem Set No. 8
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End of Document
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FAQs

What are the formulas for the sum and product of roots in quadratic equations?
The formulas for the sum and product of the roots of a quadratic equation ax² + bx + c = 0 are given by -b/a for the sum and c/a for the product. These formulas are derived from Vieta's relations, which relate the coefficients of the polynomial to sums and products of its roots. Understanding these formulas is crucial for solving quadratic equations and analyzing their roots.

How do you derive a quadratic equation from its roots?
To derive a quadratic equation from its roots r1 and r2, you can use the formula x² - (r1 + r2)x + (r1 ● r2) = 0. This formula incorporates the sum and product of the roots, allowing you to construct the quadratic equation directly. For example, if the roots are 3 and -5, the equation becomes x² - (3 + (-5))x + (3 * -5) = 0, simplifying to x² + 2x - 15 = 0.

What examples are provided for calculating the sum and product of roots?
Examples provided include quadratic equations such as x² + 5x + 4 = 0 and x² - 8x - 15 = 0. For the first equation, the sum of the roots is -5 and the product is 4. For the second equation, the sum is 8 and the product is -15. These examples illustrate how to apply the formulas for different quadratic equations.

What types of problems are included in the lesson's problem set?
The problem set includes various quadratic equations for students to practice calculating the sum and product of roots. Problems range from simple equations like n² - 5n + 6 = 0 to more complex ones such as 3x² - 2x - 1 = 0. This variety helps reinforce the concepts taught in the lesson and prepares students for real exam scenarios.

How does the lesson help students prepare for algebra exams?
This lesson equips students with the necessary skills to solve quadratic equations, which are a common topic in algebra exams. By understanding the sum and product of roots, students can tackle a wide range of problems effectively. The inclusion of examples and practice problems ensures that students can apply these concepts in various contexts, enhancing their problem-solving abilities.

What is the significance of Vieta's formulas in this lesson?
Vieta's formulas are significant in this lesson as they provide a direct relationship between the coefficients of a polynomial and the sums and products of its roots. Understanding these relationships allows students to derive quadratic equations from known roots and to analyze the properties of the roots without needing to solve the equations directly. This foundational knowledge is essential for advanced algebra and calculus.