https://file.aiquickdraw.com/mathbot/file/1725568487532nfevraj1.jpg

The image shows a graph with the function \( y = -x^2 + 4x - 3 \) and an input asking to find the solutions for \( y = 0 \) (i.e., the x-intercepts).

To solve \( -x^2 + 4x - 3 = 0 \), we can either factor or use the quadratic formula.

### Using the Quadratic Formula:
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( -x^2 + 4x - 3 = 0 \), we have:
- \( a = -1 \)
- \( b = 4 \)
- \( c = -3 \)

Substituting these into the quadratic formula:
\[
x = \frac{-4 \pm \sqrt{4^2 - 4(-1)(-3)}}{2(-1)}
\]
\[
x = \frac{-4 \pm \sqrt{16 - 12}}{-2}
\]
\[
x = \frac{-4 \pm \sqrt{4}}{-2}
\]
\[
x = \frac{-4 \pm 2}{-2}
\]
Now, we calculate the two possible values for \( x \):

1. \( x = \frac{-4 + 2}{-2} = \frac{-2}{-2} = 1 \)
2. \( x = \frac{-4 - 2}{-2} = \frac{-6}{-2} = 3 \)

Thus, the solutions are \( x = 1 \) and \( x = 3 \).

These are the points where the graph intersects the x-axis.

Let me know if you need any further clarifications!

### Here are 5 related questions:
1. How do you find the vertex of a quadratic function?
2. What does the discriminant tell you about the number of solutions to a quadratic equation?
3. How can you rewrite a quadratic function in vertex form?
4. What is the axis of symmetry of the quadratic function \( y = -x^2 + 4x - 3 \)?
5. What role does the leading coefficient play in determining the shape of a parabola?

### Tip:
When using the quadratic formula, always check the discriminant (inside the square root) first. It quickly tells you how many real solutions the equation has.

Learn how to solve the quadratic equation y = -x^2 + 4x - 3 for its x-intercepts using the quadratic formula. Follow the step-by-step solution, including calculations and graph analysis. Suitable for students in Grades 9-12.

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