Question

Write the discriminant of $x(x+4)=12$ and determine the nature of its root.

Answer 1

Step 1: Writing the given equation in standard form and comparing

The standard form of the quadratic equation is $a{x}^2+bx+c=0$.

The given quadratic equation is $x(x+4)=12$.

$\Rightarrow x^2+4x-12=0$

Here, $a=1,b=4$ and $c=-12$.

Step 2: Finding the discriminant

Using the formula for discriminant $\left(D\right)=b^2-4ac$ and substituting the values of $a,b$ and $c$ we get,

$$\begin{gathered} \Rightarrow D=4^2-4\left(1\right)(-12) \\ \Rightarrow D=16+48 \\ \Rightarrow D=64 \end{gathered}$$

Step 3: Determining the nature of the roots of the equation

We know that the value of the discriminant decides the nature of the roots of the quadratic equation.

If $D>0$, then the quadratic equation has real and unequal roots.

If $D=0$, then the quadratic equation has real and equal roots.

If $D<0$, then the quadratic equation has no real roots.

For the given quadratic equation $D>0$. So, the equation has real and unequal roots.

Therefore, the value of the discriminant is $64$ and the quadratic equation $x(x+4)=12$ has real and unequal roots.