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Solve the equation of each quadratic equation of each number 

I will solve each quadratic equation listed in the image step by step. The equations visible are:

1. \( 5x^2 - 31x - 20 \)
2. \( 7x^2 + 45x - 28 \)
3. \( 6y^2 + 24y \)
4. \( 9z^2 + 17z + 2 \)
5. \( 10a^2 + 4a - 6b^2 - 80b^2 \)

Let's proceed with factoring or solving each equation.

### 1. \( 5x^2 - 31x - 20 \)
We will factor this quadratic equation by finding two numbers that multiply to \( 5 \times (-20) = -100 \) and add to \( -31 \).

- The numbers are \( -36 \) and \( 5 \).
  
Rewriting the middle term:
\[
5x^2 - 36x + 5x - 20
\]
Factor by grouping:
\[
(5x(x - 4)) + (5(x - 4)) = (5x + 5)(x - 4)
\]
Thus, the factored form is:
\[
(5x + 5)(x - 4)
\]
Setting each factor to zero gives solutions \( x = -1 \) and \( x = 4 \).

### 2. \( 7x^2 + 45x - 28 \)
We will use the same process here, finding two numbers that multiply to \( 7 \times (-28) = -196 \) and add to \( 45 \).

- The numbers are \( 49 \) and \( -4 \).

Rewriting the middle term:
\[
7x^2 + 49x - 4x - 28
\]
Factor by grouping:
\[
7x(x + 7) - 4(x + 7) = (7x - 4)(x + 7)
\]
Thus, the factored form is:
\[
(7x - 4)(x + 7)
\]
Setting each factor to zero gives solutions \( x = \frac{4}{7} \) and \( x = -7 \).

### 3. \( 6y^2 + 24y \)
This equation can be factored by factoring out the greatest common factor:
\[
6y(y + 4)
\]
Setting each factor to zero gives solutions \( y = 0 \) and \( y = -4 \).

### 4. \( 9z^2 + 17z + 2 \)
We will factor by finding two numbers that multiply to \( 9 \times 2 = 18 \) and add to \( 17 \).

- The numbers are \( 18 \) and \( -1 \).

Rewriting the middle term:
\[
9z^2 + 18z - z + 2
\]
Factor by grouping:
\[
9z(z + 2) - 1(z + 2) = (9z - 1)(z + 2)
\]
Thus, the factored form is:
\[
(9z - 1)(z + 2)
\]
Setting each factor to zero gives solutions \( z = \frac{1}{9} \) and \( z = -2 \).

### 5. \( 10a^2 + 4a - 6b^2 - 80b^2 \)
This equation seems to have some repeated or extra terms. Clarifying the exact form is needed, but if we simplify the \( b^2 \) terms, the equation could be rewritten as:
\[
10a^2 + 4a - 86b^2
\]
This is a quadratic in both \( a \) and \( b \), but it requires clarification before proceeding with factoring.

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Would you like more details on any of these solutions, or clarifications on the last equation?

Here are 5 related questions:
1. How can you solve a quadratic equation using the quadratic formula?
2. What is the difference between factoring and using the quadratic formula?
3. How do you recognize if a quadratic equation can be factored?
4. What are other methods for solving quadratic equations besides factoring?
5. How does the discriminant help in solving quadratic equations?

**Tip:** Always check if you can factor out a greatest common factor before attempting other methods of solving quadratic equations!

Solve the equation of each quadratic equation of each number

Learn how to solve quadratic equations like 5x^2 - 31x - 20 and 7x^2 + 45x - 28 through factoring techniques. This guide provides step-by-step solutions with detailed explanations suitable for Grades 9-11 students.

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