FIND THE SUM AND PRODUCT OF ROOTS OF A QUADRATIC EQUATION

Problem 1 :

Find the sum and product of roots of the quadratic equation

x² + 5x + 6 = 0

Solution :

x² + 5x + 6 = 0

A general form of a quadratic equation ax² + bx + c = 0

a = 1, b = 5, c = 6

Sum of the roots = -b/a = -5/1 = -5

Products of the roots = c/a = 6/1 = 6

So, the sum and products of the roots are -5 and 6 respectively.

Problem 2 :

Find the sum and product of roots of the quadratic equation

x² - 4x - 10 = 0

Solution :

x² - 4x - 10 = 0

A general form of a quadratic equation ax² + bx + c = 0

a = 1, b = -4, c = -10

Sum of the roots = -b/a = -(-4)/1 = 4

Products of the roots = c/a = -10/1 = -10

So, the sum and products of the roots are 4 and 10 respectively.

Problem 3 :

Find the sum and product of roots of the quadratic equation

2x² + 6x + 8 = 0

Solution :

2x2 + 6x + 8 = 0

A general form of a quadratic equation ax2 + bx + c = 0

a = 2, b = 6, c = 8

Sum of the roots = -b/a = -6/2 = -3

Products of the roots = c/a = 8/2 = 4

So, the sum and products of the roots are -3 and 4 respectively.

Problem 4 :

Find the sum and product of roots of the quadratic equation

3x² + 5x - 9 = 0

Solution :

3x² + 5x - 9 = 0

A general form of a quadratic equation ax² + bx + c = 0

a = 3, b = 5, c = -9 

Sum of the roots = -b/a = -5/3 

Products of the roots = c/a = -9/3 = -3

So, the sum and products of the roots are -5/3 and -3 respectively.

Problem 5 :

Find the sum and product of roots of the quadratic equation

5x² - 7x - 10 = 0

Solution :

5x² - 7x - 10 = 0

A general form of a quadratic equation ax² + bx + c = 0

a = 5, b = -7, c = -10

Sum of the roots = -b/a = -(-7)/5 = 7/5 

Products of the roots = c/a = -10/5 = -2

So, the sum and products of the roots are 7/5 and -2 respectively.

Problem 6 :

Form the equation whose roots are 7and -10.

Solution :

Given, roots are 7and -10

Sum of the roots = α + β 

Product of roots = αβ

 α = 7, β = -10

x² – (α + β)x + αβ = 0

x² – (-3)x + (-70) = 0

x² + 3x - 70 = 0

So, the equation is x² + 3x - 70 = 0.

Find all values of k for which the equation has

(a) two solutions

(b) one solution and

(c) no solutions.

Problem 7 :

2x² + x + 3k = 0

Solution :

2x² + x + 3k = 0

a = 2, b = 1 and c = 3k

a) When it has two solution, the discriminant b² - 4ac > 0

1² - 4(2)(3k) > 0

1 - 24k > 0

-24k > -1

k < 1/24

b) When it has one solution, the discriminant b² - 4ac = 0

1² - 4(2)(3k) = 0

1 - 24k = 0

-24k = -1

k = 1/24

c) When it has two solution, the discriminant b² - 4ac < 0

1² - 4(2)(3k) < 0

1 - 24k < 0

-24k < -1

k > 1/24

Problem 8 :

x² − 4kx + 36 = 0

Solution :

x² − 4kx + 36 = 0

a = 1, b = -4k and c = 36

a) When it has two solutions, the discriminant b² - 4ac > 0

(-4k)² - 4(1)(36) > 0

16k² - 144 > 0

16k² > 144

k² > 144/16

k > 12/4

k > 3

b) When it has one solution, the discriminant b² - 4ac = 0

(-4k)² - 4(1)(36) = 0

16k² - 144 = 0

16k² = 144

k² = 144/16

k² = 9

k = -3 and 3

c) When it has two solution, the discriminant b² - 4ac < 0

(-4k)² - 4(1)(36) < 0

16k² - 144 < 0

16k² < 144

k² < 144/16

k² < 9

k < -3 and k < 3