To solve the equation [tex]\( x^2 - 36 = 0 \)[/tex] by factoring, we can follow these steps:
1. Identify the form of the equation:
The given equation is a quadratic equation and can be written as:
[tex]\[
x^2 - 36 = 0
\][/tex]
2. Recognize it as a difference of squares:
Notice that [tex]\( x^2 - 36 \)[/tex] is a difference of squares. The difference of squares formula is given by:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
In this particular equation, [tex]\( a = x \)[/tex] and [tex]\( b = 6 \)[/tex]. So we can rewrite the equation as:
[tex]\[
x^2 - 6^2 = 0
\][/tex]
3. Factor the difference of squares:
Apply the difference of squares formula to factor the quadratic expression:
[tex]\[
x^2 - 36 = (x - 6)(x + 6) = 0
\][/tex]
4. Set each factor equal to zero:
Setting each factor equal to zero gives us two simple linear equations to solve:
[tex]\[
x - 6 = 0 \quad \text{and} \quad x + 6 = 0
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Solve each equation separately:
[tex]\[
x - 6 = 0 \implies x = 6
\][/tex]
[tex]\[
x + 6 = 0 \implies x = -6
\][/tex]
6. List the solutions:
The solutions to the equation [tex]\( x^2 - 36 = 0 \)[/tex] are:
[tex]\[
x = -6 \quad \text{and} \quad x = 6
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] that satisfy the equation are [tex]\( x = -6 \)[/tex] and [tex]\( x = 6 \)[/tex].
