In this page framing quadratic equation1 we are going to see how to construct any quadratic equation with given roots.

**Question 1:**

Construct a quadratic equation whose two roots are -2 and -3

**Solution:**

Here two roots are -2 and -3

α = -2

β = -3

**General form of any quadratic equation x² - (α + β) x + αβ = 0**

Sum of roots (α + β) = -2 + (-3)

= - 2 - 3

= -5

Product of roots (α β) = -2(-3)

= 6

Now let us write the quadratic equation with sum and product of roots

x² - (-5) x + (6) = 0

x² + 5 x + 6 = 0

x² + x - 12 = 0

**Question 2:**

Construct a quadratic equation whose two roots are 5 and -3

**Solution:**

Here two roots are 5 and -3

α = 5

β = -3

**General form of any quadratic equation x² - (α + β) x + αβ = 0**

Sum of roots (α + β) = 5 + (-3)

= 5 - 3

= 2

Product of roots (α β) = 5(-3)

= -15

Now let us write the quadratic equation with sum and product of roots

x² -2 x + (-15) = 0

x² - 2 x - 15 = 0

**Question 3:**

Construct a quadratic equation whose two roots are 1 and 3

**Solution:**

Here two roots are 1 and 3

α = 1

β = 3

**General form of any quadratic equation x² - (α + β) x + αβ = 0**

Sum of roots (α + β) = 1 + 3

= 4

Product of roots (α β) = 1(3)

= 3

Now let us write the quadratic equation with sum and product of roots

x² - 4 x + 3 = 0