Question

How do you solve ${{x}^{2}}+6x-7=0$?

Answer 1

Hint: In this question, we are given a quadratic equation and we need to find the value of x which satisfies this equation. For this we will split the middle term method. For an equation of the form $a{{x}^{2}}+bx+c=0$ we need to find two numbers ${{n}_{1}},{{n}_{2}}$ to split the middle term (b) such that ${{n}_{1}}+{{n}_{2}}=b\text{ and }{{n}_{1}}\cdot {{n}_{2}}=ac$. After splitting we will take common terms out of the first two terms and common terms from the last two terms such that after taking common we are left with the same terms. This will help us in factoring. Then putting each factor equal to 0 we will get the value of x.

Complete step-by-step answer:
Here we are given the equation as ${{x}^{2}}+6x-7=0$. The equation is a quadratic equation (of degree 2) so we will get two values of x. Let us use the split middle term method for solving this equation. We have the equation as ${{x}^{2}}+6x-7=0$.
We know that for an equation of the form $a{{x}^{2}}+bx+c=0$ we need ${{n}_{1}},{{n}_{2}}$ such that ${{n}_{1}}+{{n}_{2}}=b\text{ and }{{n}_{1}}\cdot {{n}_{2}}=ac$. So here we have a = 1, b = 6 and c = -7.
So we need ${{n}_{1}},{{n}_{2}}$ such that ${{n}_{1}}+{{n}_{2}}=6\text{ and }{{n}_{1}}\cdot {{n}_{2}}=-7$.
As we know only two numbers which when multiplied gives 7 are 1 and 7. So if we take 7 and -1 we get 7-1 = 6 and (7)(1) = 7 that satisfies both conditions. So ${{n}_{1}}=7\text{ and }{{n}_{2}}=-1$.
Splitting the middle term in this way we get ${{x}^{2}}+\left( 7-1 \right)x-7=0\Rightarrow {{x}^{2}}+7x-x-7=0$.
Let us take x common from the first two terms and -1 from the last two terms so we have $x\left( x+7 \right)-1\left( x+7 \right)=0$.
As we can see after taking common terms we are left with (x+7) from both. So let us take (x+7) common from both terms we get $\left( x-1 \right)\left( x+7 \right)=0$.
Putting both these factors as equal to 0 we have,
(I) x-1 = 0
Adding 1 to both sides we get,
$x-1+1=0+1\Rightarrow x=1$.
(II) x+7 = 0.
Subtracting 7 from both sides we get,
\[x+7-7=0-7\Rightarrow x=-7\].
Hence the two values of x are 1 and -7 and these values will satisfy the given equation.

Note: Students should take care of the signs while selecting the number ${{n}_{1}},{{n}_{2}}$. Note that we must be left with the same factor (x+7) so that we can factorize the equation. Number of solutions is the same as that of the degree of the equation. We could also use a quadratic formula for finding the solution.