Question

How do you factor $4{{x}^{2}}+4x-3$ ?

Answer 1

Hint: For the following problem, a quadratic equation is given which is in the form of$a{{x}^{2}}+bx+c=0$. We have to find the roots or zeros for this equation. You need to know that there are only two roots in a quadratic equation. Roots are those values which satisfy the equation and result in zero when roots are placed and solved. To find the zeros, quadratic formula is used:
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

Complete step by step answer:
Now, let’s solve the problem.
As we know that the equation which has the 2 as its highest degree and is in the form of $a{{x}^{2}}+bx+c=0$ is known as quadratic equation where ‘a’ and ‘b’ are coefficients of ${{x}^{2}}$ and x respectively, and c is the constant term. It has two roots which satisfy the equation and results in zero after placing the roots in an equation. And you should know that there are basically two methods of finding roots of an equation. These two methods are middle term splitting and by using quadratic formulas. So, we will use a quadratic formula for this equation because middle term splitting is only possible if we can split the middle term into two terms and we are getting something common while grouping them in pairs as we are not able to split its middle term. Quadratic formula is:
 $\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Now, write the equation given in the question.
$\Rightarrow 4{{x}^{2}}+4x-3=0$
Here, a = 4, b = 4 and c = -3. Place all the values in quadratic formula, we will get:
$\Rightarrow x=\dfrac{-4\pm \sqrt{{{\left( 4 \right)}^{2}}-4\left( 4 \right)\left( -3 \right)}}{2\left( 4 \right)}$
On solving further:
$\Rightarrow x=\dfrac{-4\pm \sqrt{16+48}}{8}$
Solve the under root:
$\Rightarrow x=\dfrac{-4\pm \sqrt{64}}{8}$
Under root of 64 is 8:
$\Rightarrow x=\dfrac{-4\pm 8}{8}$
Solve for x:
$\Rightarrow x=\dfrac{-4+8}{8}=\dfrac{4}{8}=\dfrac{1}{2}$
$\Rightarrow x=\dfrac{-4-8}{8}=\dfrac{-12}{8}=-\dfrac{3}{2}$

$\therefore $ Roots of x are: $\dfrac{1}{2},-\dfrac{3}{2}$

Note: If you will know the values for square and square roots, there will be no need to find the LCM and check the value. So try to learn some values for square and square roots. Before applying a quadratic formula, first try to solve with the middle term splitting because it is the easiest approach for any quadratic equation.