Question

Solve the following equation by formula method.
\[{{x}^{2}}-7x+12=0\]

Answer 1

Hint: In order to find the solution of this question, we should know about the formula method of the quadratic equation. For any quadratic equation, \[a{{x}^{2}}+bx+c=0,\] we will use the formula, \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] whenever we have been asked to use the formula method or Shridhara Charya’s formula. In our question, we have the terms a = 1, b = -7 and c = 12. Now, after substituting these values in the given formula, it will be easy for us to solve for the value of x. The obtained values of x are also called roots of the equation.

Complete step-by-step solution -
In this question, we have been asked to find the solution of \[{{x}^{2}}-7x+12=0\] by the formula method. So, to solve this question, we should know about the formula method, that is, for any quadratic equation, \[a{{x}^{2}}+bx+c=0,\] the roots are given by \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}.\]
So, we can say that for the equation, \[{{x}^{2}}-7x+12=0,\] we get the value of a = 1, b = – 7 and c = 12 on comparing it with \[a{{x}^{2}}+bx+c=0.\] So, we can say the roots of the equation will be,
\[x=\dfrac{-\left( -7 \right)\pm \sqrt{{{\left( -7 \right)}^{2}}-4\left( 1 \right)\left( 12 \right)}}{2\left( 1 \right)}\]
Now, we will simplify it further to get the value of x. So, we get,
\[x=\dfrac{7\pm \sqrt{49-48}}{2}\]
And we know that it can be further written as,
\[x=\dfrac{7\pm \sqrt{1}}{2}\]
And it is as same as,
\[x=\dfrac{7\pm 1}{2}\]
Now, we will write the values of x, one with a positive sign and the other with a negative sign. So, we get,
\[x=\dfrac{7+1}{2};x=\dfrac{7-1}{2}\]
And we can further write it as,
\[x=\dfrac{8}{2};x=\dfrac{6}{2}\]
Thus, we get,
\[x=4;x=3\]
Hence, after solving the equation \[{{x}^{2}}-7x+12=0,\] we get the value of x as 4 and 3 by the formula method.

Note: We can also verify our answer by putting the obtained value of x in the given quadratic equation, if they satisfy the equation, then our answer is absolutely correct. We can also express the obtained values of x in equation form as (x-4)(x-3)=0. Expanding this, we can check if we are getting the equation given in the question or not. Also, we have to be very careful while doing calculations and writing the formula because we might make a mistake in a hurry.