Question

Write all the values of p for which the quadratic equation x^2+px+16=0 has equal roots . Find theroots of the equation so obtained

Answer 1

Answer: p=8

Since the given equation has equal roots , we can say that

》D=0

》b^2 - 4ac=0

》p^2 - 64 =0

》p^2 = 64

Therefore , p=8

So the equations now is: x^2 + 8x +16 =0

And the root of this equation is: -4

Answer 2

$\: \: \: \: \: \: \: \: \: \: \: \: {x}^{2} + px + 16 = 0$

$as \: the \: equation \: have \: equal \: roots\\so \: \: \: \: \red{\boxed { b {}^{2} - 4ac = 0}}$

$here \: \: \: \: a = 1 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: b = p \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: c = 16$

$\: \: \: \: \: \: \: \: \: \: \: \: {p}^{2} - 4(1)(16) = 0$

$\: :⟹ \: \: \: \: \: \: \: \: \: \: {p}^{2} - 64 = 0$

$\: \: :⟹ \: \: \: \: \: \: \: \: \: \: \: {p}^{2} = 64$

$\: \: \: \: :⟹ \: \: \: \: \: \: \: \: \: p = \sqrt{64}$

$\: \: \: : ⟹ \: \: \: \: \: \: \: \: \: \ \fcolorbox{blue}{pink}{\fcolorbox{green}{aqua} {p = ±8}}$

$\: \: \: \: \: \: \: \: \: p(x) = {x}^{2} + px + 16$

$when \: \: \: \: \pink{p = 8}$

$:⟹ \: \: \: {x}^{2} + 8x + 16 = 0$

$: ⟹ \: \: {x}^{2} + 4x + 4x + 16 = 0$

$: ⟹x(x + 4) + 4(x + 4) = 0$

$: ⟹ \: \: \: \: (x + 4)(x + 4) = 0$

$: ⟹ \: \: \: \: \: \: \: \: \: \: \: \boxed{x = - 4}$

$when \: \: \: \pink{p = - 8}$

$: ⟹ {x}^{2} - 8x + 16 = 0$

$: ⟹ {x}^{2} - 4x - 4x + 16 = 0$

$: ⟹x(x - 4) - 4(x - 4) = 0$

$: ⟹(x - 4)(x - 4) = 0$

$: ⟹ \: \: \: \: \: \: \: \: \boxed{x = 4}$

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