How do you solve \[2x\left( x-1 \right)=3\]?

Hint: For the given question we are given to solve the equation \[2x\left( x-1 \right)=3\]. First we have to write the equation and multiply the terms on the left hand side of the equation. Then we have to transpose all the x terms to the left hand side and all the variables to the right hand side of the equation. We will end up with a quadratic equation, which can be solved using the quadratic formula.

Complete step by step solution:
For the given equation first we have to arrange the problem as it is like the problem was given
\[2x\left( x-1 \right)=3\]
Now we have to multiply the x co-efficient and distribute the equation
\[\Rightarrow 2{{x}^{2}}-2x=3\]
Now we should send the variable to left hand side
\[\Rightarrow 2{{x}^{2}}-2x-3=0\]
And now we should use the quadratic formula
\[\Rightarrow \]\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Once in standard form, identify a, b and c from the original equation and plug them into the quadratic formula
\[\Rightarrow 2{{x}^{2}}-2x-3=0\]
a = 2
b = - 2
c = - 3
\[\Rightarrow \]\[x=\dfrac{-\left( -2 \right)\pm \sqrt{{{\left( -2 \right)}^{2}}-4\left( 2 \right)\left( -3 \right)}}{2\left( 2 \right)}\]
And then we have to evaluate the exponents
\[\Rightarrow \]\[x=\dfrac{2\pm \sqrt{{{\left( -2 \right)}^{2}}-4\left( 2 \right)\left( -3 \right)}}{2\left( 2 \right)}\]
\[\Rightarrow x=\dfrac{2\pm \sqrt{4-4\left( 2 \right)\left( -3 \right)}}{2\left( 2 \right)}\]
And then we have to multiply the numbers
\[\Rightarrow x=\dfrac{2\pm \sqrt{4+24}}{2\left( 2 \right)}\]
And then we have to add the numbers
\[\Rightarrow x=\dfrac{2\pm \sqrt{28}}{2\left( 2 \right)}\]
And then we have to factorise the numbers
\[\Rightarrow x=\dfrac{2\pm \sqrt{2\cdot 14}}{2\left( 2 \right)}\]
\[\Rightarrow x=\dfrac{2\pm \sqrt{2\cdot 2\cdot 7}}{2\left( 2 \right)}\]
And then we have to evaluate the square root
\[\Rightarrow x=\dfrac{2\pm \sqrt{2}\cdot \sqrt{2}\cdot \sqrt{7}}{2\left( 2 \right)}\]
\[\Rightarrow x=\dfrac{2\pm 2\sqrt{7}}{2\left( 2 \right)}\]
And we have to multiply the numbers
\[\Rightarrow x=\dfrac{2\pm 2\sqrt{7}}{4}\]
To solve for the unknown variable, separate into two equations: one with a plus and the other with a minus
\[\Rightarrow x=\dfrac{2+2\sqrt{7}}{4}\]
\[\Rightarrow x=\dfrac{2-2\sqrt{7}}{4}\]
Now we have to rearrange and isolate the variable to find each solution
\[\Rightarrow x=\dfrac{1+\sqrt{7}}{2}\]
\[\Rightarrow x=\dfrac{1-\sqrt{7}}{2}\]
And now we get the solution
\[\Rightarrow x=\dfrac{1\pm \sqrt{7}}{2}\]
Therefore, here is the exact solution of the problem.

Note: This type of problems cannot be solved by using factorization method. We can also check our answer by substituting the value of ‘x’ in the equation and we have to check the equation whether LHS=RHS or not.