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To form a Quadratic Equation

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☼ To form a Quadratic Equation :– A quadratic equation whose roots are $\alpha$ ₁ and $\beta$ ₁ is given by :

x² – $(\alpha_{1}+ \beta_{1})x \textup{ + } \alpha_{1} \times \beta_{1}=0$

i.e., x² – ( Sum of roots ) x + ( product of roots ) = 0

e.g. (i) Form a quadratic equation whose roots are 2 and 3.

Soln. Here, ₁ = 2 and ₁ = 3

$\therefore$ Required quadratic equation is :

x² – $(\alpha_{1}+ \beta_{1})x \textup{ + } \alpha_{1} \times \beta_{1}=0$

$\therefore$ x² – ( 2 + 3 ) x + ( 2 $\times$ 3 ) = 0

$\therefore$ x² – 5x + 6 = 0

(ii) Form a quadratic equation whose one root is 4 + $\sqrt{7}$

Soln. Here, $\alpha$ ₁ = 4 + $\sqrt{7}$ $\large \Rightarrow$ $\beta$ ₁ = 4 – $\sqrt{7}$

$\alpha_{1}+\beta_{1}$

= 4 + $\sqrt{7}$ + 4 – $\sqrt{7}$

= 8

$\therefore$ Required quadratic equation is :

x² – ( $\alpha_{1}+\beta_{1}$ ) x + $\alpha_{1}\times\beta_{1}$ = 0

$\therefore$ x² – 8x + 9 = 0

$\alpha_{1}\times\beta_{1}$

= (4 + $\sqrt{7}$ )( 4 – $\sqrt{7}$ )

= (4)²– ( $\sqrt{7}$ )²

= 9

(iii) Form a quadratic equation whose one root is 3 + 5 i

Soln. Here, $\alpha$ ₁ = 3 + 5 i $\large \Rightarrow$ $\beta_{1}$ = 3 – 5 i

$\alpha_{1}+\beta_{1}$

= 3 + 5 i + 3 – 5 i

= 6

Required quadratic equation is :

x² – ( $\alpha_{1}+\beta_{1}$ ) x + $\alpha_{1}\times\beta_{1}$ = 0

$\therefore$ x² – 6x + 34 = 0

$\alpha_{1}\times\beta_{1}$

= (3 + 5 i)( 3 – 5 i)

= ( 3 )² – (5 i)²

= 34 ( $\because$ i² = – 1 )