Question

How do you factor completely ${x^2} + 2x - 15$?

Answer 1

Hint: Here, we are required to solve the given quadratic equation and find its factored form. Thus, we will use the method of middle term splitting, and hence, by taking the terms common and simplifying further, we will be able to find the required factored form which will be the required answer.

Complete step by step solution:
Given the quadratic equation is:
${x^2} + 2x - 15$
Now, since, we are required to completely factorize this quadratic equation
Thus, we will do middle term splitting.
$ \Rightarrow {x^2} + 2x - 15 = {x^2} + 5x - 3x - 15$
Taking out the common terms from the first two and then the last two terms, we get,
$ \Rightarrow {x^2} + 2x - 15 = x\left( {x + 5} \right) - 3\left( {x + 5} \right)$
Now, taking the brackets common,
$ \Rightarrow {x^2} + 2x - 15 = \left( {x - 3} \right)\left( {x + 5} \right)$
If we equate this to 0, then, we get,
Either $x - 3 = 0$
$ \Rightarrow x = 3$
Or $x + 5 = 0$
$ \Rightarrow x = - 5$
Thus, the two possible values of $x$ are 3 and $ - 5$

And, we can factor completely ${x^2} + 2x - 15$ as $\left( {x - 3} \right)\left( {x + 5} \right)$
Thus, this is the required answer.

Note:
Since in the question, it is not mentioned that we have to use the method of middle term splitting, thus, we could have even used the method of completing the square to solve the given quadratic equation.
Given quadratic equation is:
${x^2} + 2x - 15$
Now, since, we are required to solve this question using completing the square, hence, we will write this quadratic equation as:
${\left( x \right)^2} + 2\left( x \right)\left( 1 \right) + {\left( 1 \right)^2} - {\left( 1 \right)^2} - 15 = 0$
Hence, even if this quadratic equation was not a perfect square, but we tried to make it a perfect square by adding and subtracting the square of a constant such that, we complete the square by making the identity, ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$
Hence, using this identity, we get,
${\left( {x + 1} \right)^2} - 1 - 15 = 0$
$ \Rightarrow {\left( {x + 1} \right)^2} - 16 = 0$
Adding 16 on both the sides,
$ \Rightarrow {\left( {x + 1} \right)^2} = 16$
Thus, taking square root on both sides, we get,
$\left( {x + 1} \right) = \sqrt {16} = \pm 4$
Subtracting 1 from both the sides,
$ \Rightarrow x = 4 - 1 = 3$
And
$x = - 4 - 1 = - 5$
Hence, the possible values of $x$ are $3, - 5$
Thus, we can also write this as: $\left( {x - 3} \right)\left( {x + 5} \right)$
Hence, this is the factored form of the given quadratic equation.
Therefore, this is the required answer.