Question

How do you solve the equation ${{x}^{2}}-3x-18=0$ by using a quadratic formula?

Answer 1

Hint: Here in this question, we are given a quadratic equation in the form of $a{{x}^{2}}+bx+c=0$. Therefore, we need to solve the given equation in order to find the roots or zeros. Roots or zeroes are the values which satisfy the equation. To solve a quadratic equation, we need to apply the formula in order to find the roots. Quadratic Formula is:
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

Complete step by step answer:
Now, let’s solve the question.
The equation which has the highest degree as 2, and which is in the form of $a{{x}^{2}}+bx+c=0$ is known as a quadratic equation where a and b are coefficients and c is the constant term. Roots or zeros are those values which satisfy the equation. In other words we can say that after placing these values, the equation becomes equal to zero. Common method of finding roots is splitting the middle term but in question it is mentioned to solve using quadratic formula.
So the quadratic formula is:
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Write the equation given in question.
$\Rightarrow {{x}^{2}}-3x-18=0$
Here, a = 1, b = -3, c = -18. Now place all the values in the quadratic formula:
$\Rightarrow x=\dfrac{-\left( -3 \right)\pm \sqrt{{{\left( -3 \right)}^{2}}-4\left( 1 \right)\left( -18 \right)}}{2\left( 1 \right)}$
Now, solve further:
$\Rightarrow x=\dfrac{3\pm \sqrt{9+72}}{2}$
Next is to solve the under root. We will get:
$\Rightarrow x=\dfrac{3\pm \sqrt{81}}{2}$
Under root of 81 is 9. Place 9 above:
$\Rightarrow x=\dfrac{3\pm 9}{2}$
Now, write both the values of roots:
$\therefore x=\dfrac{3+9}{2}=\dfrac{12}{2}=6$
$\therefore x=\dfrac{3-9}{2}=\dfrac{-6}{2}=-3$
The values of x are: 6, -3.
This is our final answer.

Note: If in any question, we cannot perform middle term splitting, then we have to go for this method. But in some questions both the methods work. But the easiest way is middle term splitting and if we need to verify the answer, apply a quadratic formula and check the answer. Students should remember quadratic formulas for questions related to quadratic equations.