Question

How do you write ${x^2} + 8x + 16$ in factored form?

Answer 1

Hint: The quadratic equations have two factors. We can find the factors by factoring methods where we have to split the middle term and by taking common, we can find the respective factors. The splitting is done by finding the factors such that the product is equal to the product of coefficient of ${x^2}$ and constant term while sum is the coefficient of x.

Complete step-by-step solution:
We are given a quadratic equation ${x^2} + 8x + 16 = 0$
Finding factors by splitting the middle term, in this method we have to find two numbers whose addition is the coefficient of x here is $8$ and multiplication is the constant part here is $16$. Now, these two numbers are $4,4$ which satisfies the above two conditions
Hence, we are splitting the middle term
$\Rightarrow {x^2} + 4x + 4x + 16 = 0$
Now, taking common from the first two terms and the last two terms respectively,
$\Rightarrow x(x + 4) + 4(x + 4) = 0$
Now, $(x + 4)(x + 4) = 0$
Equating the two brackets to zero. Hence, the factors of the given equation are $ - 4, - 4$

Note: We have to be very careful while selecting the two numbers. Signs should also be taken care of while selecting the numbers. The sum of the selected numbers is the coefficient of x and the product of numbers is the constant term. If we are unable to find the numbers use the formula above written in the hint part.