https://file.aiquickdraw.com/mathbot/file/1726073826181ne2017cq.jpg

The system of equations given in the image is:

1. \( y = 2x^2 + 3x - 2 \)
2. \( y = -2x^2 + 7x - 3 \)

To find the points where these two parabolas intersect, we can set the two equations equal to each other and solve for \( x \):

\[
2x^2 + 3x - 2 = -2x^2 + 7x - 3
\]

Let’s simplify and solve this equation.

### Step 1: Combine like terms
Move all terms to one side of the equation:
\[
2x^2 + 3x - 2 + 2x^2 - 7x + 3 = 0
\]
\[
4x^2 - 4x + 1 = 0
\]

### Step 2: Solve the quadratic equation
The quadratic equation is now:
\[
4x^2 - 4x + 1 = 0
\]

We can use the quadratic formula to solve this equation. The quadratic formula is:

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

For the equation \( 4x^2 - 4x + 1 = 0 \), the coefficients are:
- \( a = 4 \)
- \( b = -4 \)
- \( c = 1 \)

Now, we can compute the value of \( x \). Let me calculate this for you.The solution to the quadratic equation is \( x = 0.5 \). Since both roots are the same, there is only one solution for \( x \).

### Step 3: Find \( y \)
Now, substitute \( x = 0.5 \) into one of the original equations to find the corresponding \( y \)-value. Let’s use the first equation:

\[
y = 2x^2 + 3x - 2
\]

Substitute \( x = 0.5 \) into this equation:

\[
y = 2(0.5)^2 + 3(0.5) - 2
\]

Let me calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can still substitute \( x = 0.5 \) into the equation:

\[
y = 2(0.5)^2 + 3(0.5) - 2
\]

This simplifies to:

\[
y = 2(0.25) + 1.5 - 2 = 0.5 + 1.5 - 2 = 0
\]

Thus, the point of intersection is \( (0.5, 0) \).

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:
1. How can you graphically represent the two parabolas?
2. What does it mean when two curves intersect at only one point?
3. How can the discriminant in a quadratic equation help you understand the nature of the solutions?
4. What are some methods to solve systems of nonlinear equations?
5. How would the solution change if the quadratic equation had two distinct roots?

**Tip**: If you encounter repeated roots in a quadratic equation, it means the curves are tangent at that point!

Solve the system of equations: y = 2x^2 + 3x - 2 and y = -2x^2 + 7x - 3

Learn how to solve a system of quadratic equations by finding the intersection points of two parabolas. This problem involves using the quadratic formula and solving equations algebraically. Suitable for students in Grades 9-11.

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