Question

What is the larger solution to the equation
-2x^2+4x+4=-2x+8

Answer 1

To solve the equation \(-2x^2 + 4x + 4 = -2x + 8\), we first rearrange all terms to one side of the equation:

\[
-2x^2 + 4x + 4 + 2x - 8 = 0
\]

Combine the like terms:

\[
-2x^2 + (4x + 2x) + (4 - 8) = 0
\]

This simplifies to:

\[
-2x^2 + 6x - 4 = 0
\]

Now, we can multiply the entire equation by -1 to make the leading coefficient positive:

\[
2x^2 - 6x + 4 = 0
\]

Next, we can use the quadratic formula, where \(a = 2\), \(b = -6\), and \(c = 4\):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Calculating the discriminant:

\[
b^2 - 4ac = (-6)^2 - 4(2)(4) = 36 - 32 = 4
\]

Now substituting back into the quadratic formula:

\[
x = \frac{6 \pm \sqrt{4}}{2 \cdot 2}
\]

This simplifies to:

\[
x = \frac{6 \pm 2}{4}
\]

Now we can calculate the two possible solutions:

1. \(x = \frac{6 + 2}{4} = \frac{8}{4} = 2\)
2. \(x = \frac{6 - 2}{4} = \frac{4}{4} = 1\)

Thus, the two solutions to the equation are \(x = 2\) and \(x = 1\).

The larger solution is:

\[
\boxed{2}
\]