FIND QUADRATIC EQUATION HAVING THE GIVEN ROOTS

From each problem, find a quadratic equation with those numbers as its solutions.

Problem 1 :

2, 5

Solution :

Given, 2, 5

α = 2, β = 5

x² – (α + β)x + αβ = 0

x² – (2 + 5)x + (2 × 5) = 0

x² – 7x + 10 = 0

Problem 2 :

3, 6

Solution :

Given, 3, 6

α = 3, β = 6

x² – (α + β)x + αβ = 0

x² – (3 + 6)x + (3 × 6) = 0

x² – 9x + 18 = 0

Problem 3 :

20, 2

Solution :

Given, 20, 2

α = 20, β = 2

x² – (α + β)x + αβ = 0

x² – (20 + 2)x + (20 × 2) = 0

x² – 22x + 40 = 0

Problem 4 :

13, 1

Solution :

Given, 13, 1

α = 13, β = 1

x² – (α + β)x + αβ = 0

x² – (13 + 1)x + (13 × 1) = 0

x² – 14x + 13 = 0

Problem 5 :

4, 4

Solution :

Given, 4, 4

α = 4, β = 4

x² – (α + β)x + αβ = 0

x² – (4 + 4)x + (4 × 4) = 0

x² – 8x + 16 = 0

Problem 6 :

0, 9

Solution :

Given, 0, 9

α = 0, β = 9

x² – (α + β)x + αβ = 0

x² – (0 + 9)x + (0 × 9) = 0

x² – 9x + 0 = 0

Problem 7 :

-2, -5

Solution :

Given, -2, -5

α = -2, β = -5

x² – (α + β)x + αβ = 0

x² – (-2 + (-5))x + (-2 × -5) = 0

x² + 7x + 10 = 0

Problem 8 :

-4, 11

Solution :

Given, -4, 11

α = -4, β = 11

x² – (α + β)x + αβ = 0

x² – ((-4) + 11)x + (-4 × 11) = 0

x² – 7x - 44 = 0

Problem 9 :

3, -1

Solution :

Given, 3, -1

α = 3, β = -1

x² – (α + β)x + αβ = 0

x² – (3 + (-1))x + (3 × -1) = 0

x² – 2x - 3 = 0

Problem 10 :

3/4, 1/4

Solution :

Given, 3/4, 1/4

α = 3/4, β = 1/4

x² – (α + β)x + αβ = 0

x² – (3/4 + 1/4)x + (3/4 × 1/4) = 0

x² – (4/4)x + 3/16 = 0

Multiply 16 on each sides.

16x² – 16x + 3 = 0

Problem 11 :

5/8, 5/7

Solution :

α = 5/8

 β = 5/7

α + β = 5/8 + 5/7

= 5/8 × 7/7 + 5/7 × 8/8

= 35/56 + 40/56

= 75/56

α × β = 5/8 × 5/7

= 25/56

x² – (α + β)x + αβ = 0

x² – (75/56)x + 25/56 = 0

Multiply 56 on each sides.

56x² – 75x + 25 = 0

Problem 12 :

1/2, 1/3

Solution :

α = 1/2

 β = 1/3

α + β = 1/2 + 1/3

= 1/2 × 3/3 + 1/3 × 2/2

= 3/6 + 2/6

= 5/6

α × β = 1/2 × 1/3

= 1/6

x² – (α + β)x + αβ = 0

x² – (5/6)x + 1/6 = 0

Multiply 6 on each sides.

6x² – 5x + 1 = 0

Problem 13 :

1/2, 2/3

Solution :

α = 1/2

 β = 2/3

α + β = 1/2 + 2/3

= 1/2 × 3/3 + 2/3 × 2/2

= 3/6 + 4/6

= 7/6

α × β = 1/2 × 2/3

= 2/6

x² – (α + β)x + αβ = 0

x² – (7/6)x + 2/6 = 0

Multiply 6 on each sides.

6x² – 7x + 2 = 0