A) Solve the following quadratic equation using the perfect square trinomial pattern:

Question

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: \[ \boxed{-4} \]