Question

What is a quadratic equation? How can you solve a quadratic equation by factorization method?

Answer 1

Hint: We solve this question by explaining the concept of quadratic equations and by giving their general form which is represented as $a{{x}^{2}}+bx+c=0.$ Then we explain the method of factorization employed in order to solve a quadratic equation and we show this with the help of an example.

Complete step by step solution:
In order to answer this question, let us explain the concept of quadratic equations. A quadratic equation is a polynomial equation of second degree which means the highest power of a coefficient is two. The general form for representing a quadratic equation is given by,
$\Rightarrow a{{x}^{2}}+bx+c=0$
Here, x is the unknown variable and a, b and c are the constant coefficients.
In order to solve a quadratic equation by factorization, we need to make sure that one side of the equation is zero and the other side consists of the factors of the polynomial equation.
Now to explain the method of solving a quadratic equation, let us consider an example. Take a quadratic equation of the form ${{x}^{2}}+3x-10=0.$ This is a quadratic equation which can be represented in terms of its factors by splitting the middle term. We need two numbers such that their sum is +3 and their product is -10. The two numbers are found to be 5 and -2.
$\Rightarrow {{x}^{2}}+3x-10=0$
Splitting the middle term as 5x and -2x,
$\Rightarrow {{x}^{2}}+5x-2x-10=0$
Now taking x common out from the first two terms and -2 common from the last two terms,
$\Rightarrow x\left( x+5 \right)-2\left( x+5 \right)=0$
Now taking the term $\left( x+5 \right)$ common from both the terms,
$\Rightarrow \left( x+5 \right)\left( x-2 \right)=0$
The left-hand side of the equation consists of the factors of the quadratic equation and we find the solution to this equation by just equating each of the individual terms to 0.
$\Rightarrow \left( x+5 \right)=0$
$\Rightarrow \left( x-2 \right)=0$
Simplifying the two by subtracting 5 on both sides of the equation for the first equation and adding 2 on both sides of the second equation, we get,
$\Rightarrow x=-5\text{ or 2}$
Hence, we have shown the method of solving a quadratic equation by factorization.

Note: We need to know the basic concepts of quadratic equations to solve many problems under this topic. It is important to note that there is another method of solving quadratic equations by using the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}.$ Using this, by substituting the coefficients, we get the solution to the quadratic equations.