Question

Write the discriminant of the quadratic equation ${{\left( x+5 \right)}^{2}}=2\left( 5x-3 \right)$ .

Answer 1

Hint: We have to convert the given equation to the standard form $a{{x}^{2}}+b+c=0$ by applying algebraic identities and properties. Then, we have to compare the result with the standard form and find the values of a, b and c. Finally, we have to use the formula for discriminant which is given by $D={{b}^{2}}-4ac$ , substitute the values of a, b and c and simplify.

Complete step by step answer:
We have to write the discriminant of the quadratic equation ${{\left( x+5 \right)}^{2}}=2\left( 5x-3 \right)$ . Let us first make the given equation of the form $a{{x}^{2}}+b+c=0$ .
We know that ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ .
$\begin{align}
  & \Rightarrow {{x}^{2}}+2x\times 5+{{5}^{2}}=2\left( 5x-3 \right) \\
 & \Rightarrow {{x}^{2}}+10x+25=2\left( 5x-3 \right) \\
\end{align}$
We have to apply distributive property on the RHS.
$\Rightarrow {{x}^{2}}+10x+25=10x-6$
We have to move all the terms in the RHS to the LHS.
$\Rightarrow {{x}^{2}}+10x+25-10x+6=0$
Let us add the like terms.
$\Rightarrow {{x}^{2}}+31=0...\left( i \right)$
We know that discriminant of a quadratic equation $a{{x}^{2}}+b+c=0$ is given by
$D={{b}^{2}}-4ac$
Let us compare equation (i) with the standard form. We will get
$\begin{align}
  & a=1 \\
 & b=0 \\
 & c=31 \\
\end{align}$
Let us substitute the above values in the formula for the discriminant.
$\begin{align}
  & \Rightarrow D=0-4\times 1\times 31 \\
 & \Rightarrow D=-124 \\
\end{align}$

Hence, the discriminant of the quadratic equation ${{\left( x+5 \right)}^{2}}=2\left( 5x-3 \right)$ is -124.

Note: students have a chance of making a mistake by writing the formula for discriminant as $D={{b}^{2}}+4ac$ .The value of the discriminant can be used to find the nature of the roots of the quadratic equation. If $D>0$ , then the roots are real and unequal. If $D=0$ , then the roots are real and equal. If $D<0$ , then the roots are not real (we get a complex solution). For the given equation, we obtained $D<0$ , that is, the roots are imaginary.