Question

-8x^2+7x+49 in vertex form

Answer 1

To convert the quadratic function \(-8x^2 + 7x + 49\) into vertex form, we can use the method of completing the square.

The vertex form of a quadratic is given by:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola.

Let's go through the steps to convert \(-8x^2 + 7x + 49\) into vertex form.

1. **Factor out the coefficient of \(x^2\) from the first two terms:**

\[
-8(x^2 - \frac{7}{8}x) + 49
\]

2. **Complete the square:**

To complete the square inside the parentheses, take half of the coefficient of \(x\) (which is \(-\frac{7}{8}\)), square it, and add and subtract this value inside the parentheses.

Half of \(-\frac{7}{8}\) is \(-\frac{7}{16}\), and squaring it gives:

\[
\left(-\frac{7}{16}\right)^2 = \frac{49}{256}
\]

Now we can rewrite it as:

\[
-8\left(x^2 - \frac{7}{8}x + \frac{49}{256} - \frac{49}{256}\right) + 49
\]

This simplifies to:

\[
-8\left(\left(x - \frac{7}{16}\right)^2 - \frac{49}{256}\right) + 49
\]

3. **Distribute the \(-8\) and simplify:**

\[
-8\left(x - \frac{7}{16}\right)^2 + 8 \cdot \frac{49}{256} + 49
\]

Calculating \(8 \cdot \frac{49}{256}\):

\[
\frac{392}{256} = \frac{49}{32}
\]

Now, we have:

\[
-8\left(x - \frac{7}{16}\right)^2 + 49 + \frac{49}{32}
\]

To combine \(49\) and \(\frac{49}{32}\), convert \(49\) into a fraction with a common denominator:

\[
49 = \frac{49 \cdot 32}{32} = \frac{1568}{32}
\]

Therefore:

\[
\frac{1568}{32} + \frac{49}{32} = \frac{1617}{32}
\]

4. **Final vertex form:**

Putting it all together, we get:

\[
y = -8\left(x - \frac{7}{16}\right)^2 + \frac{1617}{32}
\]

Thus, the vertex form of the quadratic function \(-8x^2 + 7x + 49\) is:

\[
y = -8\left(x - \frac{7}{16}\right)^2 + \frac{1617}{32}
\]