Question

How do you solve ${{x}^{2}}-5x-3=0$ ?

Answer 1

Hint: We are given a quadratic equation in the form of $a{{x}^{2}}+bx+c=0$. So, we need to solve the given equation in order to find the roots or zeros. To solve a quadratic equation, we have to apply the formula to find the roots. 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 having the highest degree of 2, and which is in the form $a{{x}^{2}}+bx+c=0$ is called a quadratic equation where a and b are coefficients and c is the constant term. Common method of finding roots is splitting the middle term but in this question middle term splitting is not possible so we will approach another method in which we will be using a quadratic formula for finding the roots or zeros. In the quadratic equation, there are maximum 2 roots.
Quadratic formula is:
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Now, write the equation which is given in question:
$\Rightarrow {{x}^{2}}-5x-3=0$
Here, the value of a = 1, b = -5, c = -3
After placing all the values in the formula we will get:
$\Rightarrow x=\dfrac{-\left( -5 \right)\pm \sqrt{{{\left( -5 \right)}^{2}}-4\left( 1 \right)\left( -3 \right)}}{2\left( 1 \right)}$
Next step is to solve further:
$\Rightarrow x=\dfrac{5\pm \sqrt{25+12}}{2}$
Solve the under root. We will get:
$\Rightarrow x=\dfrac{5\pm \sqrt{37}}{2}$
Now write both the roots of this equation:
$\begin{align}
  & \therefore x=\dfrac{5+\sqrt{37}}{2} \\
 & \therefore x=\dfrac{5-\sqrt{37}}{2} \\
\end{align}$
So, this is our final answer.

Note: If in any question, we cannot do middle term splitting, then we have to go for this method. But in some questions both the methods work. So it becomes easy to find the answer. In the final step, we have to leave the answer in the roots. There is no need to solve further. Remember the quadratic formula for these types of questions.