Question

How do you solve \[{x^2} + 3x - 18 = 0\] by factoring?

Answer 1

Hint: Here we will solve the equation by using the middle term split method. Firstly we will split the middle term of the expression. Then we will take commonly from two sets of terms and get our equation in product form. Finally, we will equate the equation equal to zero and get the required solution.

Complete step-by-step answer:
The equation given to us is \[{x^2} + 3x - 18 = 0\].
First, we will split the middle term of the equation.
If we want to factorize an expression like \[a{x^2} + bx + c\] we take two numbers such that their product is equal to \[a \cdot c\] and their sum is equal to\[b\].
Let us take one number \[6\] and another number \[ - 3\] for equation \[{x^2} + 3x - 18 = 0\].
As we can see
\[6 \times - 3 = - 18 = a \cdot c\]
\[6 - 3 = 3 = b\]
So we can rewrite our equation as,
\[{x^2} + 6x - 3x - 18 = 0\]
Now taking \[x\] common in first two terms and \[ - 3\] common in last two terms we get,
\[ \Rightarrow x\left( {x + 6} \right) - 3\left( {x + 6} \right) = 0\]
Factoring out common terms, we get
\[ \Rightarrow \left( {x - 3} \right)\left( {x + 6} \right) = 0\]
Applying zero product property, we get
\[\begin{array}{l} \Rightarrow \left( {x - 3} \right) = 0\\ \Rightarrow x = 3\end{array}\]
Or
\[\begin{array}{l} \Rightarrow \left( {x + 6} \right) = 0\\ \Rightarrow x = - 6\end{array}\]
So, we got our zeroes as \[x = 3\] and \[x = - 6\].

Hence, the required factors are \[\left( {x - 3} \right)\left( {x + 6} \right)\].

Note: A quadratic equation is the equation having the variable with the highest power as two. This method is known as factoring as we find out the factors for the equation. As the highest power of the variable is two we get two factors for the equation. We find our answer by substituting the product value equal to zero because the equation has to have at least one value equal to zero for the equation to be equal to zero.