Solve y = -4x² + 9: Finding X-Axis Intersection Points

Q: Why do I set y = 0 to find x-axis intersections?

The x-axis is where y = 0! Every point on the x-axis has a y-coordinate of zero. So to find where our curve crosses the x-axis, we substitute y = 0 into our equation.

Q: How do I know there are two intersection points?

When we solve $ x^2 = \frac{9}{4} $, taking the square root gives us two values: positive and negative. This means our parabola crosses the x-axis at two different points.

Q: What does the ± symbol mean exactly?

The ± symbol means "plus or minus". So $ x = \pm\frac{3}{2} $ gives us two solutions: $ x = \frac{3}{2} $ and $ x = -\frac{3}{2} $.

Q: Why are the y-coordinates both 0 in the final answer?

Because we're finding x-axis intersections! On the x-axis, every point has y = 0. Our answers $ (-\frac{3}{2}, 0) $ and $ (\frac{3}{2}, 0) $ show where the curve touches the x-axis.

Q: Can I check my answer by graphing?

Absolutely! The equation $ y = -4x^2 + 9 $ is a downward-opening parabola. You should see it cross the x-axis at exactly $ x = -1.5 $ and $ x = 1.5 $.

Quadratic Functions with X-Axis Intersections

Determine the points of intersection of the function

$9-y=4x^2$

With the X

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the intersection point of the function with the X-axis

00:03 At the intersection point with the X-axis, Y equals 0

00:07 Substitute Y=0 in our equation and solve to find the intersection point

00:18 Isolate X

00:26 Extract the root

00:35 Remember when extracting a root there are 2 solutions (positive and negative)

00:38 Calculate root 9 and root 4

00:48 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Determine the points of intersection of the function

$9-y=4x^2$

With the X

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Set $y = 0$ in the given equation $9 - y = 4x^2$ to get the simplified equation $9 = 4x^2$ .
Step 2: Rearrange the equation to isolate $x^2$ , yielding $4x^2 = 9$ .
Step 3: Solve for $x^2$ by dividing each side by 4, resulting in $x^2 = \frac{9}{4}$ .
Step 4: Take the square root of both sides to find $x = \pm \frac{3}{2}$ .

Therefore, the points of intersection with the x-axis are $(-\frac{3}{2}, 0)$ and $(\frac{3}{2}, 0)$ .

Referring to the provided choices, this correctly corresponds to choice 3.

Final Answer

$(-\frac{3}{2},0),(\frac{3}{2},0)$

Key Points to Remember

Essential concepts to master this topic

Intersection Rule: Set y = 0 to find x-intercepts
Technique: From $4x^2 = 9$ , divide by 4 to get $x^2 = \frac{9}{4}$
Check: Substitute back: $4(\pm\frac{3}{2})^2 = 4 \cdot \frac{9}{4} = 9$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the ± when taking square roots
Don't write just x = 3/2 after taking the square root = missing half the answer! Square roots always give both positive and negative solutions. Always write x = ±3/2 to get both intersection points.

Practice Quiz

Test your knowledge with interactive questions

Which chart represents the function $$ y=x^2-9 $$ ?

FAQ

Everything you need to know about this question

Why do I set y = 0 to find x-axis intersections?

The x-axis is where y = 0! Every point on the x-axis has a y-coordinate of zero. So to find where our curve crosses the x-axis, we substitute y = 0 into our equation.

How do I know there are two intersection points?

When we solve $x^2 = \frac{9}{4}$ , taking the square root gives us two values: positive and negative. This means our parabola crosses the x-axis at two different points.

What does the ± symbol mean exactly?

The ± symbol means "plus or minus". So $x = \pm\frac{3}{2}$ gives us two solutions: $x = \frac{3}{2}$ and $x = -\frac{3}{2}$ .

Why are the y-coordinates both 0 in the final answer?

Because we're finding x-axis intersections! On the x-axis, every point has y = 0. Our answers $(-\frac{3}{2}, 0)$ and $(\frac{3}{2}, 0)$ show where the curve touches the x-axis.

Can I check my answer by graphing?

Absolutely! The equation $y = -4x^2 + 9$ is a downward-opening parabola. You should see it cross the x-axis at exactly $x = -1.5$ and $x = 1.5$ .

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