Question

A) Solve the following quadratic equation using the perfect square trinomial pattern:
49x
2 = − 392x − 784. (1 point)
x = 784
x = − 784
x = − 4
x = 4

Answer 1

To solve the quadratic equation \( 49x^2 = -392x - 784 \) using the perfect square trinomial pattern, let's first rewrite the equation in standard form:

\[
49x^2 + 392x + 784 = 0
\]

Next, we can see if we can express the left-hand side as a perfect square. Notice that we can factor out 49 from the first two terms:

\[
49(x^2 + 8x) + 784 = 0
\]

Now, let's complete the square for the expression inside the parentheses:

1. Take half of the coefficient of \( x \) (which is 8) and square it:
\[
\left(\frac{8}{2}\right)^2 = 4^2 = 16
\]

2. Add and subtract this square inside the parentheses:
\[
49(x^2 + 8x + 16 - 16) + 784 = 0
\]
\[
49((x + 4)^2 - 16) + 784 = 0
\]

3. Distribute the 49:
\[
49(x + 4)^2 - 784 + 784 = 0
\]
\[
49(x + 4)^2 = 0
\]

Now, divide both sides by 49:
\[
(x + 4)^2 = 0
\]

Taking the square root of both sides gives:
\[
x + 4 = 0
\]

Therefore, solving for \( x \) gives:
\[
x = -4
\]

So, the solution to the equation is:
\[
\boxed{-4}
\]