# FRAMING A QUADRATIC EQUATION WITH GIVEN ROOTS

Framing a Quadratic Equation with Given Roots :

When two roots of a quadratic equation are given , the formula to form the quadratic equation is given by

x² - (sum of the roots)x + product of the roots = 0

If  and ᵦ be the two roots of a quadratic equation are given , then the formula to form the quadratic equation is given by

x² - (α + β) x + αβ = 0

Example 1 :

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

Solution :

Roots are α  =  -2 and β  =  -3

x² - (α + β) x + αβ = 0

 α + β  =  -2 + (-3)=  - 2 - 3=  -5 (α β)  =  -2(-3)  =  6

x2 - (-5) x + (6)  =  0

x2 + 5 x + 6  =  0

Example 2 :

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

Solution :

Roots are α  =  5 and β  =  -3

 α + β  =  5 + (-3)  =  5 - 3  =  2 α β  =  5(-3)  =  -15

x² -2 x + (-15)  =  0

x² - 2 x - 15  =  0

Example 3 :

Construct a quadratic equation whose two roots are 1 and 3

Solution :

Given roots are α  =  1 and β  =  3

 α + β = 1 + 3  =  4 α β  =  1(3)  = 3

x² - 4 x + 3  =  0

Example 4 :

Construct a quadratic equation whose two roots are 24 and -3

Solution :

The given roots are α = 24 and β = -3

 α + β  =  24 + (-3)  =  24 - 3  =  21 α β  =  24(-3)  = -72

x² - 21 x + (-72)  =  0

x² - 21 x - 72  =  0

Example 5 :

Construct a quadratic equation whose two roots are -1 and -5

Solution :

The given roots are α  =  -1 and β  =  -5

 α + β  =  -1 + (-5)  =  -1 - 5  = -6 α β  =  -1(-5)  =  5

x² - (-6) x + 5  =  0

x² + 6 x + 5  =  0

Example 6 :

Construct a quadratic equation whose two roots are -7 and 5

Solution :

The given roots are α  =  -7 and β  =  5

 α + β  =  -7 + 5  =  -2 α β  =  -7(5)  = -35

x² - (-2) x + (-35)  =  0

x² + 2 x - 35  =  0 After having gone through the stuff given above, we hope that the students would have understood how to frame a quadratic equation with given roots.

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