Question

How do you factor and solve ${x^2} + 8x + 15 = 0$ ?

Answer 1

Hint: We will apply a middle term splitting method as it is easily applicable here i.e. we will split the middle term, here the middle term is $8x$. After that we will find the factors of the given equation i.e. ${x^2} + 8x + 15 = 0$.

Complete Step by Step Solution:
The given equation is ${x^2} + 8x + 15 = 0$
Now, multiply the coefficient of ${x^2}$ with a coefficient of ${x^0}$, we get 1 x 15 =15
Positive factors of 15 are 1, 3, 5 and 15
As a product of the coefficient of ${x^2}$ and coefficient of ${x^0}$ is 15, therefore we will add the factors of 15 in such a way that the sum will become the coefficient of $x$
Now, we will add 5 and 3
$\Rightarrow {{x}^{2}}+8x+15=0$
$ \Rightarrow {x^2} + \left( {5 + 3} \right)x + 15 = 0$
$ \Rightarrow {x^2} + 5x + 3x + 15 = 0$
Now, we will take x common from the first two terms and 3 from the last two terms i.e.
$ \Rightarrow x\left( {x + 5} \right) + 3\left( {x + 5} \right) = 0$
$ \Rightarrow \left( {x + 3} \right)\left( {x + 5} \right) = 0$
Hence, factors of ${x^2} + 8x + 15 = 0$ are $\left( {x + 3} \right)$ and $\left( {x + 5} \right)$
Now, either $\left( {x + 3} \right) = 0$ or $\left( {x + 5} \right) = 0$

So, we got two values of x i.e. $x = - 3, - 5$

Note:
There is an alternative method to solve the given equation in which we use Quadratic Formula i.e. $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ where $a$ is coefficient of ${x^2}$ , $b$ is coefficient of $x$ and $c$ is coefficient of ${x^0}$
Quadratic Formula is $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
The given equation is ${x^2} + 8x + 15 = 0$
Here $a = 1$ , $b = 8$ and $c = 15$
Hence, $x = \dfrac{{ - 8 \pm \sqrt {{8^2} - 4\left( 1 \right)\left( {15} \right)} }}{{2\left( 1 \right)}}$
$ \Rightarrow x = \dfrac{{ - 8 \pm \sqrt {64 - 60} }}{2}$
On further simplification,
$ \Rightarrow x = \dfrac{{ - 8 \pm \sqrt 4 }}{2}$
Now, one value of $x = \dfrac{{ - 8 + 2}}{2}$ and another value of $x = \dfrac{{ - 8 - 2}}{2}$ i.e.
$x = \dfrac{{ - 6}}{2}$ and $x = \dfrac{{ - 10}}{2}$
$ \Rightarrow x = - 3, - 5$
Hence, factors of given equation are $\left( {x + 3} \right)$ and $\left( {x + 5} \right)$.