Let's solve the quadratic equation [tex]\( x^2 + 14x + 49 = 0 \)[/tex] step by step.
1. Identify the coefficients: For the equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 14 \)[/tex]
- [tex]\( c = 49 \)[/tex]
2. Check if it's a perfect square: We can rewrite the quadratic expression [tex]\( x^2 + 14x + 49 \)[/tex] to see if it is a perfect square trinomial. We notice that:
[tex]\[
(x + 7)^2 = x^2 + 2 \cdot 7 \cdot x + 7^2 = x^2 + 14x + 49
\][/tex]
3. Rewrite the equation: Since [tex]\( x^2 + 14x + 49 = (x + 7)^2 \)[/tex], the original equation becomes:
[tex]\[
(x + 7)^2 = 0
\][/tex]
4. Solve for [tex]\( x \)[/tex]: To solve [tex]\((x + 7)^2 = 0\)[/tex], we take the square root of both sides:
[tex]\[
x + 7 = 0
\][/tex]
5. Isolate [tex]\( x \)[/tex]: Subtract 7 from both sides:
[tex]\[
x = -7
\][/tex]
Therefore, the solution to the equation [tex]\( x^2 + 14x + 49 = 0 \)[/tex] is:
[tex]\[
x = -7
\][/tex]
