If $- 4$ is a root of the quadratic equation $x^{2} + p x - 4 = 0$ and the quadratic equation $x^{2} + p x + k = 0$ has equal roots, find the value of $k$ .

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Solution

Since $- 4$ is a root of the equation $x^{2} + p x - 4 = 0$

$∴ {(- 4)}^{2} + p \times (- 4) - 4 = 0 \Rightarrow 16 - 4 p - 4 = 0 \Rightarrow 4 p = 12 \Rightarrow p = 3$

The equation $x^{2} + p x + k = 0$ has equal roots

$∴$ Discriminant, $D = b^{2} - 4 a c = 0$

$\Rightarrow p^{2} - 4 k = 0 \Rightarrow k = \frac{9}{4}$

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If −4 is a root of the quadratic equation x2+px−4=0 and the quadratic equation x2+px+k=0 has equal roots, find the value of k.

Since −4 is a root of the equation x2+px−4=0∴(−4)2+p×(−4)−4=0⇒16−4p−4=0⇒4p=12⇒p=3The equation x2+px+k=0 has equal roots∴ Discriminant, D=b2−4ac=0⇒p2−4k=0⇒k=94

If $- 4$ is a root of the quadratic equation $x^{2} + p x - 4 = 0$ and the quadratic equation $x^{2} + p x + k = 0$ has equal roots, find the value of $k$ .