To solve the quadratic equation:

\[
y^2 + 8y = 20
\]

### Step 1: Bring all terms to one side of the equation.
Subtract 20 from both sides to set the equation to zero:

\[
y^2 + 8y - 20 = 0
\]

### Step 2: Solve the quadratic equation.

#### Option 1: Factoring
First, try to factor the quadratic equation. We need to find two numbers that multiply to \(-20\) and add to \(8\). However, the numbers that fit these criteria don't exist, so this equation cannot be factored easily.

#### Option 2: Quadratic Formula
Since factoring doesn't work, we'll use the quadratic formula:

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

For the equation \(y^2 + 8y - 20 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 8\)
- \(c = -20\)

Plugging in these values:

\[
y = \frac{-8 \pm \sqrt{8^2 - 4(1)(-20)}}{2(1)}
\]

\[
y = \frac{-8 \pm \sqrt{64 + 80}}{2}
\]

\[
y = \frac{-8 \pm \sqrt{144}}{2}
\]

\[
y = \frac{-8 \pm 12}{2}
\]

### Step 3: Calculate the two possible values for \(y\).
1. \(y = \frac{-8 + 12}{2} = \frac{4}{2} = 2\)
2. \(y = \frac{-8 - 12}{2} = \frac{-20}{2} = -10\)

### Final Answer:
The solutions are:
\[
y = 2 \quad \text{or} \quad y = -10
\]

Would you like more details on any step or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would you solve the quadratic equation if the coefficients were different?
2. What is the significance of the discriminant in the quadratic formula?
3. Can you solve a quadratic equation by completing the square?
4. How do the roots of the equation change if you shift the graph vertically?
5. How do you factor quadratic equations that can be factored?

**Tip:** When using the quadratic formula, always check the discriminant (\(b^2 - 4ac\)) first. If it's positive, you have two real solutions; if it's zero, you have one real solution; and if it's negative, you have complex solutions.

Learn how to solve the quadratic equation y^2 + 8y = 20 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic equation formula and the quadratic formula theorem. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation