Factor by completing the square to solve the given equation. Enter your solution(s) as reduced fractions. If there are more than one solutions, separate them by a comma. 10q^2-31q=-15

We are given the equation:

\[
10q^2 - 31q = -15
\]

### Step 1: Move the constant to the other side
First, let's add 15 to both sides of the equation:

\[
10q^2 - 31q + 15 = 0
\]

### Step 2: Divide by the leading coefficient
To simplify, divide the entire equation by 10 to make the coefficient of \(q^2\) equal to 1:

\[
q^2 - \frac{31}{10}q + \frac{15}{10} = 0
\]

Simplify the constant term:

\[
q^2 - \frac{31}{10}q + \frac{3}{2} = 0
\]

### Step 3: Move the constant term to the other side
Next, move the constant term to the other side:

\[
q^2 - \frac{31}{10}q = -\frac{3}{2}
\]

### Step 4: Complete the square
To complete the square, take half of the coefficient of \(q\), square it, and add it to both sides.

The coefficient of \(q\) is \(-\frac{31}{10}\), so half of this is:

\[
\frac{-31}{10} \times \frac{1}{2} = \frac{-31}{20}
\]

Now, square \(\frac{-31}{20}\):

\[
\left(\frac{-31}{20}\right)^2 = \frac{961}{400}
\]

Add \(\frac{961}{400}\) to both sides:

\[
q^2 - \frac{31}{10}q + \frac{961}{400} = -\frac{3}{2} + \frac{961}{400}
\]

### Step 5: Simplify the right-hand side
Convert \(-\frac{3}{2}\) to a fraction with denominator 400:

\[
-\frac{3}{2} = \frac{-600}{400}
\]

Now add \(\frac{-600}{400}\) and \(\frac{961}{400}\):

\[
\frac{-600}{400} + \frac{961}{400} = \frac{361}{400}
\]

### Step 6: Factor the left-hand side
Now, the left-hand side is a perfect square trinomial:

\[
\left(q - \frac{31}{20}\right)^2 = \frac{361}{400}
\]

### Step 7: Solve for \(q\)
Take the square root of both sides:

\[
q - \frac{31}{20} = \pm \frac{\sqrt{361}}{\sqrt{400}} = \pm \frac{19}{20}
\]

### Step 8: Solve for \(q\)
Now, solve for \(q\) by adding \(\frac{31}{20}\) to both sides:

\[
q = \frac{31}{20} \pm \frac{19}{20}
\]

Thus, the two solutions are:

\[
q = \frac{31 + 19}{20} = \frac{50}{20} = \frac{5}{2}
\]
\[
q = \frac{31 - 19}{20} = \frac{12}{20} = \frac{3}{5}
\]

### Final Answer:
The solutions are:

\[
q = \frac{5}{2}, \frac{3}{5}
\]

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. What is the general process for completing the square in a quadratic equation?
2. How do you factor a perfect square trinomial?
3. Can you always solve quadratic equations by completing the square?
4. How does completing the square relate to the quadratic formula?
5. What are other methods to solve quadratic equations, besides completing the square?

**Tip:** When completing the square, always remember to balance the equation by adding the same value to both sides!

Learn how to solve the quadratic equation 10q^2 - 31q + 15 = 0 by completing the square. This detailed step-by-step solution covers core algebraic concepts and demonstrates the process of transforming a quadratic equation into a perfect square trinomial. Suitable for students in Grades 9-12.

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