The equation given in the image is:

$2x - 3 = -5x^2$

Let's solve this quadratic equation step by step.

Step 1: Rewrite the equation in standard quadratic form

We want to move all terms to one side to set the equation to 0:

$5x^2 + 2x - 3 = 0$

Step 2: Use the quadratic formula

The quadratic formula is:

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

For our equation $5x^2 + 2x - 3 = 0$, we have:

• $a = 5$
• $b = 2$
• $c = -3$

Step 3: Calculate the discriminant

$\Delta = b^2 - 4ac = 2^2 - 4(5)(-3) = 4 + 60 = 64$

Step 4: Solve for $x$

Now that we know the discriminant is 64, we can apply the quadratic formula:

$x = \frac{-2 \pm \sqrt{64}}{2(5)} = \frac{-2 \pm 8}{10}$

Thus, we have two solutions:

1. $x = \frac{-2 + 8}{10} = \frac{6}{10} = \frac{3}{5}$
2. $x = \frac{-2 - 8}{10} = \frac{-10}{10} = -1$

Final answer:

The two roots are $x = \frac{3}{5}$ and $x = -1$.

Thus, the correct answers are:

• A. $x = \frac{3}{5}$
• B. $x = -1$

Would you like further details or clarifications?

Here are five related questions you can explore:

1. How do you derive the quadratic formula?
2. What does the discriminant tell you about the nature of the roots?
3. Can you solve the same quadratic equation by factoring?
4. How can we graph the quadratic equation and interpret the roots?
5. What happens when the discriminant is negative?

Tip: Always check the discriminant first when solving a quadratic equation—it tells you whether the roots are real or complex!