Question

How do you solve $12=2{{x}^{2}}-5x$?

Answer 1

Hint: In this problem we need to solve the given equation which is a quadratic equation. We know that to solve a quadratic equation there are several methods like completing square, factorization and using the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. First, we will rearrange the given equation in the form of $a{{x}^{2}}+bx+c=0$ and write the value of $a$, $b$, $c$. After getting the values of $a$, $b$, $c$ we will use the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ and simplify the equation to get the required solution of the given equation.

Complete step by step answer:
Given equation, $12=2{{x}^{2}}-5x$.
Rearranging the terms in the above equation to convert it in the form of $a{{x}^{2}}+bx+c=0$. Then we will get
$\begin{align}
  & 12=2{{x}^{2}}-5x \\
 & \Rightarrow 2{{x}^{2}}-5x-12=0 \\
\end{align}$
Comparing the above equation with $a{{x}^{2}}+bx+c=0$, then we will get
$a=2$, $b=-5$, $c=-12$.
We know that the solution for the equation $a{{x}^{2}}+bx+c=0$ is
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substituting the all the values in the above equation, then we will have
$x=\dfrac{-\left( -5 \right)\pm \sqrt{{{\left( -5 \right)}^{2}}-4\left( 2 \right)\left( -12 \right)}}{2\left( 2 \right)}$
When we multiply a negative sign with a negative sign, we will get a positive sign. Then we will get
$\begin{align}
  & x=\dfrac{5\pm \sqrt{25+96}}{4} \\
 & \Rightarrow x=\dfrac{5\pm \sqrt{121}}{4} \\
\end{align}$
We know that the value of $\sqrt{121}$ is $11$, then we will get
$\Rightarrow x=\dfrac{5\pm 11}{4}$
From the above equation we can write
$x=\dfrac{5+11}{4}$ or $x=\dfrac{5-11}{4}$
Simplifying the above equation, then we will get
$\begin{align}
  & \Rightarrow x=\dfrac{16}{4} \\
 & \Rightarrow x=4 \\
\end{align}$ or $\begin{align}
  & \Rightarrow x=\dfrac{-6}{4} \\
 & \Rightarrow x=-\dfrac{3}{2} \\
\end{align}$

Hence the solution of the given equation $12=2{{x}^{2}}-5x$ is $x=4,-\dfrac{3}{2}$.

Note: We can also find the solution of the given equation by plotting a graph of the given equation on a coordinate system. We can observe the graph of the above equation as

In the above graph also, we can observe the roots of the given equation as $x=4,-\dfrac{3}{2}$.