Use the quadratic formula to solve. Express your answer in simplest form.
20, w, squared, plus, 13, w, plus, 2, equals, 0
20w 
2
 +13w+2=
0

We are given the quadratic equation:

\[
20w^2 + 13w + 2 = 0
\]

To solve using the quadratic formula, recall that for any quadratic equation in the form \( ax^2 + bx + c = 0 \), the solution is given by:

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

For the equation \( 20w^2 + 13w + 2 = 0 \), the coefficients are:

- \( a = 20 \)
- \( b = 13 \)
- \( c = 2 \)

### Step 1: Calculate the discriminant

The discriminant is \( b^2 - 4ac \).

\[
\text{Discriminant} = 13^2 - 4(20)(2) = 169 - 160 = 9
\]

### Step 2: Apply the quadratic formula

Now, substitute the values of \( a \), \( b \), and the discriminant into the quadratic formula:

\[
w = \frac{-13 \pm \sqrt{9}}{2(20)} = \frac{-13 \pm 3}{40}
\]

### Step 3: Solve for \( w \)

Now, calculate both solutions for \( w \):

1. \( w_1 = \frac{-13 + 3}{40} = \frac{-10}{40} = \frac{-1}{4} \)
2. \( w_2 = \frac{-13 - 3}{40} = \frac{-16}{40} = \frac{-2}{5} \)

### Final Answer:

The solutions are:

\[
w = \frac{-1}{4} \quad \text{or} \quad w = \frac{-2}{5}
\]

Would you like more details on any of the steps?

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Here are 5 related questions:

1. What happens if the discriminant is negative?
2. Can the quadratic formula be applied to equations that aren't in standard form?
3. How would you solve a quadratic equation with complex roots?
4. What are the real-life applications of quadratic equations?
5. How do you factor quadratics that have integer solutions?

**Tip:** Always check the discriminant before solving a quadratic—it tells you how many real solutions there are!

Use the quadratic formula to solve. Express your answer in simplest form. 20w^2 + 13w + 2 = 0

This page provides a detailed solution to the quadratic equation 20w^2 + 13w + 2 = 0 using the quadratic formula. Learn how to calculate the discriminant and find the simplest form of the solution. Suitable for students in Grades 9-10, covering key algebra concepts like solving quadratics and using the quadratic formula.

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