Question

Solve the equation $ - 4{x^2} + 8x = - 3$ and find the value of x?

Answer 1

Hint: We will be multiplying both sides by minus sign to make the coefficient of ${x^2}$ positive. We will try to convert our one side in the form of a perfect square by adding or subtracting some constant on both sides, so that it will not affect the equation. Then find the value of x.$ - 4{x^2} + 8x = - 3$

Complete step-by-step solution:
We have given $ - 4{x^2} + 8x = - 3$ we have to find the value of x.
We will multiply both side by minus sign, we get
$ \Rightarrow 4{x^2} - 8x = 3$
We have added 4 on both sides to make left side a perfect square
$ \Rightarrow 4{x^2} - 8x + 4 = 7$
We have divided the whole equation by 4.
$ \Rightarrow {x^2} - 2x + 1 = \dfrac{7}{4}$
We will convert it into square form.
$ \Rightarrow {(x - 1)^2} = \dfrac{7}{4}$
We know that the square term contains both positive and negative values.
$ \Rightarrow (x - 1) = \pm \sqrt {\dfrac{7}{4}} $
The value of x in the equation $ - 4{x^2} - 8x = - 3$ are $x = 1 + \sqrt {\dfrac{7}{4}} $ and $x = 1 - \sqrt {\dfrac{7}{4}} $ .

Note: We will first analyse the question and try to find the way it will be solved easily. Some questions can be solved using mid-term split, perfect squaring, factorization etc. to approach which method it will come with practice. Some questions may be solved by many methods but there will always be an easier one to solve.