## Relations Between Roots Solution4

The page relations between roots solution4 is containing solution of some practice questions from the worksheet relationship between roots and coefficients.

(5) If α and β are the roots of 2 x² - 3 x - 5 = 0, form a quadratic equation whose roots are α² and β².

Solution:

General form of quadratic equation whose roots are α and β

x² - (α + β) x + α β = 0

by comparing the given equation with general form of quadratic equation we get a = 2  b = -3 and c = -5

Sum of the roots α + β = -b/a

= -(-3)/2

= 3/2

Product of roots α β = c/a

= -5/2

here α = α²  and β = β²

General form of quadratic equation whose roots are α² and β²

x² - (α² + β²) x + α² β² = 0

x² - (α² + β²) x + β)² = 0

α² + β² = (α + β)² - 2 α β

= (3/2)² - 2 (-5/2)

= (9/4) + 5

= (9 + 20)/4

= 29/4

x² - (α² + β²) x + β)² = 0

x² - (29/4) x + (-5/2)² = 0

x² - (29/4) x + (25/4) = 0

(4 x² - 29 x + 25)/4 = 0

4 x² - 29 x + 25 = 0

Therefore the required quadratic equation is 4 x² - 29 x + 25 = 0.

(6) If α and β are the roots of x² - 3 x + 2 = 0, form a  quadratic equation whose roots are -α and -β.

Solution:

General form of quadratic equation whose roots are α and β

x² - (α + β) x + α β = 0

by comparing the given equation with general form of quadratic equation we get a = 1  b = -3 and c = 2

Sum of the roots α + β = -b/a

= -(-3)/1

= 3

Product of roots α β = c/a

= 2/1

= 2

here α = - α  and β = - β

General form of quadratic equation whose roots are α² and β²

x² - ( -β) x + (-α) (-β) = 0

x² + (α + β) x + α β = 0

x² - (3) x + 2 = 0

Therefore the required quadratic equations is  x² -3 x + 2 = 0.

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