Question

How do you solve $ 2{{x}^{2}}+4x-30=0 $ ?

Answer 1

Hint: We can solve the given quadratic equation by using the process of factorization. First of all, we will consider the given equation and using the process of factorization, and find the factors of the equation. Later we will equate them to zero using the zero theorem and derive the value of $ x $ .

Complete step by step answer:
From the given question the equation is, $ 2{{x}^{2}}+4x-30=0 $
Now we will perform factorization. Factorization is the process of finding factors of any polynomial.
Factorization of any quadratic equation is done by choosing two numbers whose sum is equal to the coefficient of $ x $ when the coefficient of $ {{x}^{2}} $ is $ 1 $ and their product should be equal to the constant term.
As all the coefficients are even divide both sides of the equation by $ 2 $
By dividing the both sides of the equation by $ 2 $ , we will get the below quadratic equation, $ {{x}^{2}}+2x-15=0 $
Now we can split the constant $ -15 $ into the product of the two numbers and also sum of the two numbers must be equal to the coefficient of $ x $ that is $ 2 $ in this equation.
 $ -15=5\times -3 $
We can split $ -15 $ into the product of $ 5\text{ and -3} $
Sum of $ 5\text{ and -3} $ is $ \text{2} $ which is equal to the coefficient of $ x $
Now equation can be written as,
 $ {{x}^{2}}-3x+5x-15=0 $
 $ \Rightarrow x\left( x-3 \right)+5\left( x-3 \right)=0 $
 $ \Rightarrow \left( x+5 \right)\left( x-3 \right)=0 $
By using the zero theorem,
 $ x=-5\text{ and x=3} $ are the solutions for the given quadratic equation.

Note:
We can also answer this question using the formulae for finding the zeroes of any quadratic equation $ a{{x}^{2}}+bx+c $ given by $ \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} $ . For the equation $ 2{{x}^{2}}+4x-30=0\Rightarrow 2\left( {{x}^{2}}+2x-15=0 \right) $ the zeroes are given by $ \dfrac{-2\pm \sqrt{4-4\left( 1 \right)\left( -15 \right)}}{2}=\dfrac{-2\pm \sqrt{4+60}}{2}=\dfrac{-2\pm \sqrt{64}}{2}=\dfrac{-2\pm 8}{2}=-5,3 $ are the solutions of the given equation.