https://file.aiquickdraw.com/mathbot/file/172922736067824x2bjz5.jpg

The given equation is:

\[
20y^2 + 7y - 3 = 0
\]

This is a quadratic equation, and we can solve it using the quadratic formula:

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

Where \(a = 20\), \(b = 7\), and \(c = -3\).

### Step 1: Compute the discriminant
\[
\Delta = b^2 - 4ac = 7^2 - 4(20)(-3) = 49 + 240 = 289
\]

### Step 2: Use the quadratic formula
Now substitute the values into the quadratic formula:
\[
y = \frac{-7 \pm \sqrt{289}}{2(20)} = \frac{-7 \pm 17}{40}
\]

### Step 3: Find the two solutions
1. For the positive root:
\[
y_1 = \frac{-7 + 17}{40} = \frac{10}{40} = \frac{1}{4}
\]

2. For the negative root:
\[
y_2 = \frac{-7 - 17}{40} = \frac{-24}{40} = \frac{-3}{5}
\]

### Final Answer:
The solutions are \(y = \frac{1}{4}, -\frac{3}{5}\).

Let me know if you want more details or further clarification.

### Related Questions:
1. How can the discriminant help determine the number of solutions in a quadratic equation?
2. What are the steps for solving a quadratic equation using factoring?
3. How would the solution change if the discriminant were negative?
4. Can you explain why the quadratic formula always gives two solutions?
5. What are other methods to solve quadratic equations besides using the quadratic formula?

### Tip:
When using the quadratic formula, always check the discriminant first to determine the nature of the solutions—real or complex.

Learn how to solve the quadratic equation 20y^2 + 7y - 3 using the quadratic formula. This example covers fundamental algebraic concepts and the application of the quadratic formula. Ideal for students in Grades 8-10.

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