Question

Solve the following equation by factorization method. ${x^2} + 6x = 27$

Answer 1

Hint: In this question, we are given an equation which we have been asked to factorize. But first, bring the equation in the format $a{x^2} + bx + c = 0$. Then, factorize the given equation using splitting the middle term. Find two such factors of -27 such that when they are added or subtracted, you will get 6. Then, take x common and make factors. This will give you the answer.

Complete step-by-step solution:
We are given an equation ${x^2} + 6x = 27$ and we have been asked to factorize it. But, if you observe carefully, you will see that the given equation is not in the form of our general equation $a{x^2} + bx + c = 0$. So, our first step is to bring the equation in the form of our general equation.
$ \Rightarrow {x^2} + 6x = 27$ (given)
Shifting the constant term,
$ \Rightarrow {x^2} + 6x - 27 = 0$
Now, we have our equation in the form of $a{x^2} + bx + c = 0$.
Next step is to find two factors of -27 in such a way that when those factors are added or subtracted, we get 6. The two such factors can be 9 and -3.
Let us put them in the equation.
$ \Rightarrow {x^2} + 9x - 3x - 27 = 0$
Taking x common,
$ \Rightarrow x\left( {x + 9} \right) - 3\left( {x + 9} \right)$
Now we have,
$ \Rightarrow \left( {x - 3} \right)\left( {x + 9} \right)$

Therefore, the factors of ${x^2} + 6x = 27$ are $\left( {x - 3} \right)\left( {x + 9} \right)$.

Note: If you find it difficult to find the factors, you can use the quadratic formula to find the factors. This formula will give you the values of x. using them you can find the factors.
The formula is - $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
On comparing, we know that $a = 1,b = 6,c = - 27$.
Putting all the values, we will get,
$ \Rightarrow x = \dfrac{{ - 6 \pm \sqrt {{6^2} - 4\left( { - 27} \right)} }}{2}$
Simplifying the equation,
$ \Rightarrow x = \dfrac{{ - 6 \pm \sqrt {36 + 108} }}{2}$
$ \Rightarrow x = \dfrac{{ - 6 \pm \sqrt {144} }}{2}$
Simplifying it further,
$ \Rightarrow x = \dfrac{{ - 6 + 12}}{2},\dfrac{{ - 6 - 12}}{2}$
$ \Rightarrow x = 3, - 9$
Now we have,
$ \Rightarrow x = 3,x = - 9$
Shifting the constants on the other side,
$ \Rightarrow \left( {x - 3} \right)\left( {x + 9} \right) = 0$
Hence, these are the factors.