Question

Solve $3{{\left( 2x-1 \right)}^{2}}+4\left( 2x-1 \right)-4=0$

Answer 1

Hint: In this problem we need to solve the given equation. We can observe that the given equation is a quadratic equation. First, we will apply the algebraic formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ to expand the term ${{\left( 2x-1 \right)}^{2}}$. Now we will apply the distribution law of multiplication to remove all the parenthesis in the given equation. After that we will simplify the obtained equation by using some mathematical operations. Now we will get the equation as $a{{x}^{2}}+bx+c=0$. Compare the obtained equation with $a{{x}^{2}}+bx+c=0$ and use the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ to find the solution of the given equation.

Complete step by step solution:
Given equation is $3{{\left( 2x-1 \right)}^{2}}+4\left( 2x-1 \right)-4=0$.
Applying the algebraic formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ to expand the term ${{\left( 2x-1 \right)}^{2}}$, then the above equation is modified as
$3\left[ {{\left( 2x \right)}^{2}}-2\left( 2x \right)\left( 1 \right)+{{1}^{2}} \right]+4\left( 2x-1 \right)-4=0$
Simplifying the above equation, then we will get
$3\left( 4{{x}^{2}}-4x+1 \right)+4\left( 2x-1 \right)-4=0$
Applying the distribution law of multiplication to remove the parentheses in the above equation, then we will have
$12{{x}^{2}}-12x+3+8x-4-4=0$
Using mathematical operations to simplify the above equation, then we will get
$12{{x}^{2}}-4x-5=0$
Comparing the above equation with $a{{x}^{2}}+bx+c=0$, then we will get
$a=12$, $b=-4$, $c=-5$
Now the solution of the above equation will be
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substituting the values $a=12$, $b=-4$, $c=-5$ in the above equation, then we will get
$x=\dfrac{-\left( -4 \right)\pm \sqrt{{{\left( -4 \right)}^{2}}-4\left( 12 \right)\left( -5 \right)}}{2\left( 12 \right)}$
Simplifying the above equation by using mathematical operations, then we will have
$\begin{align}
  & x=\dfrac{4\pm \sqrt{16+240}}{24} \\
 & \Rightarrow x=\dfrac{4\pm \sqrt{256}}{24} \\
\end{align}$
We know that the value of $\sqrt{256}$ is $16$. Substituting this value in the above equation, then we will get
$x=\dfrac{4\pm 16}{24}$
Simplifying the above equation by using mathematical operations, then we will have
$\begin{align}
  & x=\dfrac{4+16}{24}\text{ or }\dfrac{4-16}{24} \\
 & \Rightarrow x=\dfrac{20}{24}\text{ or }\dfrac{-12}{24} \\
 & \Rightarrow x=\dfrac{5}{6}\ \text{or }-\dfrac{1}{2} \\
\end{align}$

Hence the solution for the given equation $3{{\left( 2x-1 \right)}^{2}}+4\left( 2x-1 \right)-4=0$ is $x=\dfrac{5}{6},-\dfrac{1}{2}$.

Note: We can also solve this problem by using the graphical method. In this method we will plot the graph of the given equation and find the roots of the given equation. The graph of the given equation will be

From the above graph also, we can say that the solution of the given equation as $x=\dfrac{5}{6},-\dfrac{1}{2}$.