Question

How do we solve ${x^2} + 6x - 7 = 0$ using the quadratic formula?

Answer 1

To solve this question, we have to go through the quadratic formula to find the roots of the given equation. Then substitute the coordinates of the equation into the quadratic formula. That’s how, we will get the roots or values of $x$ .

Complete step by step solution:
The given equation is:
${x^2} + 6x - 7 = 0$
As we know, the general form of the quadratic equation: $a{x^2} + bx + c$ .
So the coordinates of the above equation as a, b and c are $1$ , $6$ and $- 7$ respectively.
Now,
To solve this question, means to find the roots of the given equation or to find the values of $x$ .
So, for finding roots, we will use Quadratic formula:-
$\dfrac{{ - b \pm \sqrt {{b^2} - 4(a.c)} }}{{2a}}$
Substitute the values $a = 1$ , $b = 6$ and $c = - 7$ into the above quadratic formula and solve for $x$ :
$\therefore \dfrac{{ - 6 \pm \sqrt {( - 6) - 4(1. - 7)} }}{{2.1}}$
$= \dfrac{{ - 6 \pm 8}}{2}$
Now, we can also write the above value of $x$ as $\dfrac{{ - 6 + 8}}{2}$ and $\dfrac{{ - 6 - 8}}{2}$ ,
So, the roots of given equation are:
$\;\dfrac{{ - 6 + 8}}{2},\dfrac{{ - 6 - 8}}{2}$
$= - \dfrac{{14}}{2},\dfrac{2}{2}$
$= - 7,1$
Therefore, the values of $x$ or the roots of the given equation are $- 7$ and $1$ .
$x = - 7,1$

Note:- To crosscheck, whether the found roots or the values of $x$ are correct or not. We have to put the values of $x$ in the given quadratic equation. And, after putting values, if we will find the L.H.S is equals to R.H.S, then the found roots are correct:
${( - 7)^2} + 6.( - 7) - 7 = 0$ or ${1^2} + 6.1 - 7 = 0$
$\Rightarrow 49 - 42 - 7 = 0$ or $\Rightarrow 1 + 6 - 7 = 0$
$\Rightarrow 0 = 0$ or $\Rightarrow 0 = 0$
Hence, $L.H.S = R.H.S$ , the found roots are correct.