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111 LearnersLast updated on September 10, 2025
Sum and Product of Roots
To find the sum and product of roots, use this formula for any equation like ax2 + bx + c = 0 Sum of roots = - ba Product of roots = ca
Sum and Product of Roots - Examples
Problem 1
Find the sum and product of roots x2 - 7x + 10 = 0
Sum = 7 and Product = 10
Explanation
In this equation, x2 - 7x + 10 = 0 let's consider a = 1, b = -7 and c = 10
Sum of roots = - -7/1 = 7
Product of roots = 10/1 = 10
Problem 2
x2 - 6x + 9 = 0
Sum = 6 and product = 9
Explanation
Here, a = 1, b = -6 and c = 9
Use the standard formula
Sum of roots = - b/a = - -6/1 = 6
Product of roots = c/a = 9/1 = 9
Problem 3
For 2x2 - 4x + 1 = 0, find the sum and product of roots.
sum = 2 and product = 0.5
Explanation
a = 2, b = -4 and c = 1
Sum of roots = - b/a = - 4/2 = 2
Product of roots = c/a = 1/2 = 0.5
Problem 4
Find π and π½, π + π½, and π Γ π½ in the roots of the equation x2 - 7x + 12 = 0. Use these to create a new equation with roots 1/ π and 1/π½
12x2 - 7x + 1 = 0
Explanation
Here, a = 1, b = -7, and c = 12
π + π½ = - ba = - -71 = 7
π × π½ = ca = 121 = 12
New roots are 1 π and 1π½
Then find the sum and product of the new roots
Sum= 1 π + 1π½ = + = 712
Product = 1 π × 1π½ = 1 = 112
Finally, use the standard formula
x2 - (sum)x + product = 0
Then the equation becomes
x2 - 712 x + 112 = 0
Multiply through by 12 to remove fractions:
12x2 - 7x + 1 = 0
Problem 5
x2 + 5x + 5 = 0. Let Ξ± and Ξ² be the roots of this equation, and find a new equation whose roots are 1 π and 1π½
5x2 + 5x + 1 = 0
Explanation
π + π½ = −5
αβ = 5
Sum= 1 π + 1π½ = + = 55 = -1
Product = 1 π × 1π½ = 1 = 15
Form the new equation
x2 -(sum)x + product = 0
x2 - (-1)x + 15 = 0
x2 + x + 15 = 0
Multiply the equation by 5 to eliminate the denominator
5x2 + 5x + 1 = 0
Problem 6
Find the quadratic equation whose roots are π + π½ and Ξ±Ξ², where Ξ±, Ξ² are roots of 2x2 - 3x + 5 = 0.
4x2 - 16x + 15 = 0
Explanation
π + π½ = 32
αβ = 52
Now form the new roots
Sum= 3 2 + 52 = 82 = 4
Product = 3 2 × 52 = 154
Form the new quadratic equation
x2 -(sum) x + product = 0
x 2 -4x + 152 = 0
Multiply the terms by 4 to eliminate the fraction
4x2 - 16x + 15 = 0
Problem 7
Find the quadratic equation whose roots are the square of the roots of x2 -5x + 6 = 0
x2 - 13x + 36 = 0
Explanation
x2 -5x + 6 = 0 which becomes (x -2) (x -3) = 0
The roots are 2 and 3
The squares are 4 and 9
Which becomes (x -4) (x -9) = x2 - 13x + 36
x2 - 13x + 36 = 0
Problem 8
Find the sum and product of roots 6x2 + x - 12 = 0
Sum = - 16 and Product = - 12
Explanation
Here a = 6, b = 1 and c = -12
Sum of roots = - 16
Product of roots = - 121 = - 12
Problem 9
2x2 + 3x + 1 = 0
sum = - 32 and product = 12
Explanation
Here, a = 2, b = 3, and c = 1
Sum = - 32
Product = 12
Problem 10
5x2 - 15x + 50 = 0
Sum = 3, and Product = 10
Explanation
Here a = 5, b = 15, and c = 50
Sum of roots = - ba = - 155 = 3
Product of roots = ca = 505 = 10
Problem 11
xΒ² + 12x + 36 = 0
sum = -12, and product = 36
Explanation
Here, a = 1, b = 12, and c = 36
Sum of roots = - ba = - 121 = -12
Product of roots = ca = 361 = 36
Problem 12
x2 -8x + 15 = 0
sum = 8, and product = 15
Explanation
Here, a = 1, b = -8, and c = 15
Sum of roots = - ba = - -81 = 8
Product of roots = ca = 151 = 15
Problem 13
x2 - x - 20 = 0
sum = 1, and product = -20
Explanation
Here, a = 1, b = -1, and c = -20
Sum of roots = - ba = - -11 = 1
Product of roots = ca = -201 = -20
Problem 14
The roots of x2 + px + 6 = 0 have a sum of 5, find p and the product.
sum = -5, and product = 6
Explanation
Here a = 1, b = p, and c = 6
Sum of roots = - ba = - p1 = -p. Given sum = 5, -p = 5, so p = -5
Product = ca = 61 = 6
Problem 15
Find the sum and product of the equation 7x2 - 2x - 14 = 0
sum = 27, and product = -2
Explanation
Here, a = 7, b = -2, and c = -14
Sum of roots = - ba = - -27 = 27
Product of roots = ca = -147 = -2
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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