Determine Intersection Points of y=(x-2)(x+3) with the X-axis

Q: Why do we set y = 0 to find x-intercepts?

The x-intercepts are points where the graph touches or crosses the x-axis. On the x-axis, the y-coordinate is always zero, so we set y = 0 to find these special points.

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

If you have something like $ y = x^2 + x - 6 $, you'd need to factor it first to get $ y = (x-2)(x+3) $, then apply the zero-product property.

Q: Can a quadratic function have more than 2 x-intercepts?

No! A quadratic function can have at most 2 x-intercepts, exactly 1 x-intercept (if it just touches the x-axis), or no real x-intercepts (if it doesn't cross the x-axis at all).

Q: How do I remember which coordinate is which in the final answer?

Remember that x-intercepts have the form (x, 0) - the x-value you solved for goes first, and y is always 0. So x = -3 gives you (-3, 0), and x = 2 gives you (2, 0).

Q: What does the zero-product property actually mean?

The zero-product property says: "If two numbers multiply to give zero, then at least one of them must be zero." So if $ (x-2)(x+3) = 0 $, then either $ x-2 = 0 $ or $ x+3 = 0 $ (or both).

X-Intercepts with Factored Quadratic Functions

Determine the points of intersection of the function

$y=(x-2)(x+3)$

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

00:03 At the intersection point with the X-axis, the Y value must = 0

00:07 Substitute Y = 0 and solve to find X values

00:13 Find what makes each factor in the product zero

00:16 This is one solution

00:26 This is the second solution

00:32 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

$y=(x-2)(x+3)$

With the X

Step-by-step solution

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

Step 1: Recognize that to find the intersection with the x-axis, we set $y = 0$ for the function $y = (x-2)(x+3)$ .
Step 2: Solve the equation $(x-2)(x+3) = 0$ .
Step 3: Use the zero-product property, which states that if a product equals zero, then at least one of the factors must be zero. Thus:

$x-2 = 0$ or $x+3 = 0$

Step 4: Solve each equation:

For $x-2 = 0$ , we add 2 to both sides, yielding $x = 2$ .
For $x+3 = 0$ , we subtract 3 from both sides, yielding $x = -3$ .

Step 5: These x-values represent the points of intersection with the x-axis, or the x-intercepts.

Thus, the points of intersection of the function
$y = (x-2)(x+3)$ with the x-axis are the coordinates $(-3,0)$ and $(2,0)$ .

Therefore, the solution to the problem is the points $(-3,0),(2,0)$ .

Final Answer

$(-3,0),(2,0)$

Key Points to Remember

Essential concepts to master this topic

Rule: Set y = 0 to find where graph crosses x-axis
Technique: Use zero-product property: if (x-2)(x+3) = 0, then x-2 = 0 or x+3 = 0
Check: Substitute back: (2-2)(2+3) = 0×5 = 0 and (-3-2)(-3+3) = (-5)×0 = 0 ✓

Common Mistakes

Avoid these frequent errors

Setting each factor equal to y instead of zero
Don't solve x-2 = y and x+3 = y when finding x-intercepts = gives wrong coordinates! This confuses the intercept concept. Always set the entire function equal to zero: (x-2)(x+3) = 0.

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 we set y = 0 to find x-intercepts?

The x-intercepts are points where the graph touches or crosses the x-axis. On the x-axis, the y-coordinate is always zero, so we set y = 0 to find these special points.

What if the function wasn't already factored?

If you have something like $y = x^2 + x - 6$ , you'd need to factor it first to get $y = (x-2)(x+3)$ , then apply the zero-product property.

Can a quadratic function have more than 2 x-intercepts?

No! A quadratic function can have at most 2 x-intercepts, exactly 1 x-intercept (if it just touches the x-axis), or no real x-intercepts (if it doesn't cross the x-axis at all).

How do I remember which coordinate is which in the final answer?

Remember that x-intercepts have the form (x, 0) - the x-value you solved for goes first, and y is always 0. So x = -3 gives you (-3, 0), and x = 2 gives you (2, 0).

What does the zero-product property actually mean?

The zero-product property says: "If two numbers multiply to give zero, then at least one of them must be zero." So if $(x-2)(x+3) = 0$ , then either $x-2 = 0$ or $x+3 = 0$ (or both).

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