Question

If $-4$ is a root of the quadratic equation$x^2+px–4=0$ and the quadratic equation $x^2+px+k=0$ has equal roots. The value of $k$ is

• A. $\frac{9}{4}$
• B. $\frac{4}{9}$
• C. $3$
• D. None of these

Answer 1

The correct option is A

$\frac{9}{4}$

Explanation for the correct option

Step 1: Solve for the value of $p$

Given that $-4$ is a root of the quadratic equation$x^2+px–4=0$

$\therefore$ $-4$ solves the equation $x^2+px–4=0$

$\begin{array}{crcl}\Rightarrow & {\left(-4\right)}^2+\left(-4\right)p–4& =& 0\\ \Rightarrow & 16-4& =& 4p\\ \Rightarrow & 12& =& 4p\\ \Rightarrow & p& =& \frac{12}{4}\\ \Rightarrow & p& =& 3\end{array}$

Step 2: Solve for the value of $k$

Given $x^2+px+k=0$ has equal roots.

On substituting the value of $p$we get, $x^2+3x+k=0$

Given that $x^2+3x+k=0$ has equal roots

We know that if a quadratic equation $a{x}^2+bx+c=0$ has equal roots then the discriminant $b^2–4ac=0$

$\begin{array}{crcc}\Rightarrow & 3^2–4\times 1\times k& =& 0\\ \Rightarrow & 9-4k& =& 0\\ \Rightarrow & 4k& =& 9\\ \Rightarrow & k& =& \frac{9}{4}\end{array}$

Hence option(A) is correct i.e. $\frac{9}{4}$