Calculate the Intersection Points of the Quadratic Curve y=(x-3)(x+3) and the X-axis

Q: Why do the x-intercepts have y-coordinates of 0?

By definition, x-intercepts are points where the graph crosses the x-axis. Since the x-axis has equation $ y = 0 $, all x-intercepts have the form $ (x, 0) $.

Q: What if the function wasn't already factored?

You'd need to factor the quadratic first or use the quadratic formula. The factored form $ (x-3)(x+3) $ makes finding x-intercepts much easier using the zero-product property.

Q: How do I remember which sign to use when solving each factor?

Always set each factor equal to zero. For $ x-3 = 0 $, add 3 to get $ x = 3 $. For $ x+3 = 0 $, subtract 3 to get $ x = -3 $.

Q: Can I expand the function first and then solve?

You could expand to get $ y = x^2 - 9 $, then solve $ x^2 - 9 = 0 $, but using the already factored form is much faster and less prone to errors!

Q: What does this quadratic look like when graphed?

This is a parabola opening upward that crosses the x-axis at $ (-3, 0) $ and $ (3, 0) $. The vertex is at $ (0, -9) $ since it's symmetric between the x-intercepts.

X-intercepts with Factored Form

Determine the points of intersection of the function

$y=(x-3)(x+3)$

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 with the X-axis

00:04 At the intersection point with the X-axis, the Y value must equal 0

00:09 Substitute Y=0 and solve to find the appropriate X values

00:17 Find what makes each factor in the product zero

00:20 This is one solution

00:24 This is the second solution

00:28 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

$y=(x-3)(x+3)$

With the X

Step-by-step solution

To determine the points of intersection of the function $y=(x-3)(x+3)$ with the x-axis, we need to set $y$ to zero and solve for $x$ .

Follow these steps:

Step 1: Set the function equal to zero: $(x-3)(x+3) = 0$ .
Step 2: Apply the zero-product property, solving each factor for zero:

For $x-3=0$ :

$x = 3$

For $x+3=0$ :

$x = -3$

Thus, the points of intersection of the function with the x-axis, or the x-intercepts, are $(-3, 0)$ and $(3, 0)$ .

Therefore, the solution to the problem, confirming x-intercepts, is $(-3, 0), (3, 0)$ .

Final Answer

$(-3,0),(3,0)$

Key Points to Remember

Essential concepts to master this topic

Zero-Product Property: If $(x-a)(x+b) = 0$ , then $x = a$ or $x = -b$
Set Function to Zero: Replace y with 0 to get $(x-3)(x+3) = 0$
Verify Solution: Check that $(-3-3)(-3+3) = (-6)(0) = 0$ ✓

Common Mistakes

Avoid these frequent errors

Solving factors incorrectly
Don't set $x-3 = 3$ and $x+3 = -3$ = wrong solutions (6,-6)! This gives the factor values, not the x-values. Always solve $x-3 = 0$ to get $x = 3$ and $x+3 = 0$ to get $x = -3$ .

Practice Quiz

Test your knowledge with interactive questions

The following function has been graphed below:

$$ f(x)=-x^2+5x+6 $$

Calculate points A and B.

FAQ

Everything you need to know about this question

Why do the x-intercepts have y-coordinates of 0?

By definition, x-intercepts are points where the graph crosses the x-axis. Since the x-axis has equation $y = 0$ , all x-intercepts have the form $(x, 0)$ .

What if the function wasn't already factored?

You'd need to factor the quadratic first or use the quadratic formula. The factored form $(x-3)(x+3)$ makes finding x-intercepts much easier using the zero-product property.

How do I remember which sign to use when solving each factor?

Always set each factor equal to zero. For $x-3 = 0$ , add 3 to get $x = 3$ . For $x+3 = 0$ , subtract 3 to get $x = -3$ .

Can I expand the function first and then solve?

You could expand to get $y = x^2 - 9$ , then solve $x^2 - 9 = 0$ , but using the already factored form is much faster and less prone to errors!

What does this quadratic look like when graphed?

This is a parabola opening upward that crosses the x-axis at $(-3, 0)$ and $(3, 0)$ . The vertex is at $(0, -9)$ since it's symmetric between the x-intercepts.

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Calculate the Intersection Points of the Quadratic Curve y=(x-3)(x+3) and the X-axis

X-intercepts with Factored Form

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do the x-intercepts have y-coordinates of 0?

What if the function wasn't already factored?

How do I remember which sign to use when solving each factor?

Can I expand the function first and then solve?

What does this quadratic look like when graphed?

🌟 Unlock Your Math Potential

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