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Use the quadratic formula to solve. Express your answer in simplest form.
22, q, squared, plus, 16, q, minus, 5, equals, 6, q, squared
22q
2
+16q−5=
6q
2

Answer
Attempt 1 out of 5

Additional Solution
No Solution
q, equalsq=

Answer 1

To solve the equation \( 22q^2 + 16q - 5 = 6q^2 \) using the quadratic formula, we need to first rearrange the equation into standard form, which is of the form \( ax^2 + bx + c = 0 \).

Starting with the given equation:

\[
22q^2 + 16q - 5 = 6q^2
\]

We can move all terms to one side:

\[
22q^2 - 6q^2 + 16q - 5 = 0
\]

Combining like terms gives:

\[
(22 - 6)q^2 + 16q - 5 = 0
\]

This simplifies to:

\[
16q^2 + 16q - 5 = 0
\]

Now we identify \( a = 16 \), \( b = 16 \), and \( c = -5 \).

Next, we will apply the quadratic formula:

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

First, we calculate the discriminant:

\[
b^2 - 4ac = 16^2 - 4 \cdot 16 \cdot (-5)
\]

Calculating \( 16^2 \):

\[
256
\]

Calculating \( -4 \cdot 16 \cdot -5 \):

\[
= 320
\]

Now, combine these results:

\[
b^2 - 4ac = 256 + 320 = 576
\]

Calculate the square root of the discriminant:

\[
\sqrt{576} = 24
\]

Now substituting back into the quadratic formula:

\[
q = \frac{-16 \pm 24}{2 \cdot 16}
\]

This simplifies to:

\[
q = \frac{-16 \pm 24}{32}
\]

We have two potential solutions:

1.

\[
q = \frac{-16 + 24}{32} = \frac{8}{32} = \frac{1}{4}
\]

2.

\[
q = \frac{-16 - 24}{32} = \frac{-40}{32} = -\frac{5}{4}
\]

Thus, the solutions in simplest form are:

\[
q = \frac{1}{4} \quad \text{and} \quad q = -\frac{5}{4}
\]