Solve the following equation by completing the square: [6p^2 + 7p - 20 = 0]

We are given the quadratic equation:

\[
6p^2 + 7p - 20 = 0
\]

### Step 1: Divide the equation by the coefficient of \( p^2 \) (6)
We need to simplify the equation by dividing through by 6 to make the coefficient of \( p^2 \) equal to 1:

\[
p^2 + \frac{7}{6}p - \frac{20}{6} = 0
\]

This simplifies to:

\[
p^2 + \frac{7}{6}p - \frac{10}{3} = 0
\]

### Step 2: Move the constant term to the other side
Now, move the constant term \( -\frac{10}{3} \) to the right-hand side:

\[
p^2 + \frac{7}{6}p = \frac{10}{3}
\]

### Step 3: Complete the square
To complete the square, take half of the coefficient of \( p \) (which is \( \frac{7}{6} \)), square it, and add it to both sides of the equation. First, calculate half of \( \frac{7}{6} \):

\[
\frac{1}{2} \times \frac{7}{6} = \frac{7}{12}
\]

Now, square \( \frac{7}{12} \):

\[
\left(\frac{7}{12}\right)^2 = \frac{49}{144}
\]

Add \( \frac{49}{144} \) to both sides of the equation:

\[
p^2 + \frac{7}{6}p + \frac{49}{144} = \frac{10}{3} + \frac{49}{144}
\]

### Step 4: Simplify the right-hand side
Now simplify the right-hand side by converting \( \frac{10}{3} \) to a fraction with a denominator of 144:

\[
\frac{10}{3} = \frac{480}{144}
\]

So the equation becomes:

\[
p^2 + \frac{7}{6}p + \frac{49}{144} = \frac{480}{144} + \frac{49}{144}
\]

\[
p^2 + \frac{7}{6}p + \frac{49}{144} = \frac{529}{144}
\]

### Step 5: Write the left-hand side as a perfect square
The left-hand side is now a perfect square trinomial, so we can write it as:

\[
\left(p + \frac{7}{12}\right)^2 = \frac{529}{144}
\]

### Step 6: Solve for \( p \)
Take the square root of both sides:

\[
p + \frac{7}{12} = \pm \frac{\sqrt{529}}{\sqrt{144}}
\]

\[
p + \frac{7}{12} = \pm \frac{23}{12}
\]

### Step 7: Solve for \( p \)
Now, solve for \( p \) by subtracting \( \frac{7}{12} \) from both sides:

\[
p = -\frac{7}{12} \pm \frac{23}{12}
\]

This gives two solutions:

1. \( p = \frac{-7 + 23}{12} = \frac{16}{12} = \frac{4}{3} \)
2. \( p = \frac{-7 - 23}{12} = \frac{-30}{12} = -\frac{5}{2} \)

### Final Solution:
The solutions to the equation are:

\[
p = \frac{4}{3} \quad \text{or} \quad p = -\frac{5}{2}
\]

Would you like further details or explanations?

Here are 5 related questions:
1. How do you apply the method of completing the square to any quadratic equation?
2. What is the significance of completing the square in solving quadratics?
3. Can all quadratic equations be solved by completing the square?
4. How do the solutions compare when solving using the quadratic formula?
5. Why is it important to convert fractions to a common denominator when solving?

**Tip**: Completing the square is especially useful when you're working on quadratic equations that don't factor easily and can also help in graphing parabolas.

Discover how to solve the quadratic equation 6p^2 + 7p - 20 = 0 using the method of completing the square. This step-by-step guide covers essential algebraic concepts and formulas, making it ideal for students in Grades 9-10.

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