To solve the quadratic equation [tex]\(49x^2 - 64 = 0\)[/tex] by factoring, follow these detailed steps:
1. Recognize the equation as a difference of squares:
Notice that [tex]\(49x^2\)[/tex] and [tex]\(64\)[/tex] are both perfect squares. Specifically,
[tex]\[
49x^2 = (7x)^2 \quad \text{and} \quad 64 = 8^2
\][/tex]
Therefore, we rewrite the equation in the form of a difference of squares:
[tex]\[
49x^2 - 64 = (7x)^2 - 8^2
\][/tex]
2. Factor the difference of squares:
The difference of squares [tex]\(a^2 - b^2\)[/tex] can be factored into [tex]\((a - b)(a + b)\)[/tex]. Applying this to our equation, we get:
[tex]\[
(7x)^2 - 8^2 = (7x - 8)(7x + 8)
\][/tex]
Thus, the equation [tex]\(49x^2 - 64 = 0\)[/tex] becomes:
[tex]\[
(7x - 8)(7x + 8) = 0
\][/tex]
3. Set each factor to zero and solve for [tex]\(x\)[/tex]:
To find the values of [tex]\(x\)[/tex], set each factor equal to zero and solve:
[tex]\[
7x - 8 = 0 \quad \Rightarrow \quad 7x = 8 \quad \Rightarrow \quad x = \frac{8}{7}
\][/tex]
and
[tex]\[
7x + 8 = 0 \quad \Rightarrow \quad 7x = -8 \quad \Rightarrow \quad x = -\frac{8}{7}
\][/tex]
4. List the solutions:
The solutions to the quadratic equation are:
[tex]\[
x = \frac{8}{7} \quad \text{and} \quad x = -\frac{8}{7}
\][/tex]
Therefore, the correct answer is:
c. [tex]\(x = -\frac{8}{7}\)[/tex] and [tex]\(x = \frac{8}{7}\)[/tex]
