Question

How do you solve $\left( 2x-7 \right)\left( x+1 \right)=6x-19$?

Answer 1

Hint: First multiply two terms present on the left side of the equation. Then bring all the remaining terms to the left side of the equation to form a quadratic equation. Solve the quadratic equation by doing necessary calculation, to find the value of ‘x’.

Complete step by step solution:
Solving the relational equation means, we have to find the value of ‘x’, for which the equation gets satisfied.
The equation we have $\left( 2x-7 \right)\left( x+1 \right)=6x-19$
Multiplying two terms present on the left side of the equation, we get
$\begin{align}
  & \left( 2x-7 \right)\left( x+1 \right)=6x-19 \\
 & \Rightarrow 2{{x}^{2}}+2x-7x-7=6x-19 \\
 & \Rightarrow 2{{x}^{2}}-5x-7=6x-19 \\
\end{align}$
To form a quadratic equation, we have to bring all the terms to the left side of the equation.
So, bringing $6x-19$, to the left side of the equation we get
$\begin{align}
  & \Rightarrow 2{{x}^{2}}-5x-6x-7+19=0 \\
 & \Rightarrow 2{{x}^{2}}-11x+12=0 \\
\end{align}$
Solving the quadratic equation: If we have an equation of the form $a{{x}^{2}}+bx+c=0$, then we can solve the equation using the quadratic formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.
Considering our equation $2{{x}^{2}}-11x+12=0$
By comparison $a=2$, $b=-11$ and $c=12$
So, ‘x’ can be determined using the quadratic formula as
\[\begin{align}
  & x=\dfrac{11\pm \sqrt{{{\left( -11 \right)}^{2}}-4\cdot 2\cdot 12}}{2\cdot 2} \\
 & \Rightarrow x=\dfrac{11\pm \sqrt{121-96}}{4} \\
 & \Rightarrow x=\dfrac{11\pm \sqrt{25}}{4} \\
 & \Rightarrow x=\dfrac{11\pm 5}{4} \\
\end{align}\]
Either
$x=\dfrac{11+5}{4}=\dfrac{16}{4}=4$
Or,
$x=\dfrac{11-5}{4}=\dfrac{6}{4}=\dfrac{3}{2}$
Hence, the solution of $\left( 2x-7 \right)\left( x+1 \right)=6x-19$ is $x=4\text{ or }\dfrac{3}{2}$.
This is the required solution of the given question.

Note: Forming the quadratic equation should be the first approach for solving this question. After simplifying the given equation, since we are getting a second degree equation, so by solving we are getting two values of ‘x’.