Solve the Quadratic Equation: Find Intersection Points of y=(x-2)(x+4)

Q: Why do I set the function equal to zero to find x-intercepts?

X-intercepts are points where the graph crosses the x-axis. At these points, the y-coordinate is always 0. So you solve $ (x-2)(x+4) = 0 $ to find where y = 0.

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

You'd need to factor it first or use other methods like the quadratic formula. The factored form $ (x-2)(x+4) $ makes finding x-intercepts much easier!

Q: How do I remember which coordinate is which?

Think alphabetically: x comes before y, so x-intercepts are written as (x-value, 0). The y-coordinate is always 0 for x-intercepts.

Q: Can I expand the function first before solving?

You could expand to get $ x^2 + 2x - 8 = 0 $ and solve, but it's much faster to use the zero product property with the factored form!

X-Intercepts with Factored Quadratics

Determine the points of intersection of the function

$y=(x-2)(x+4)$

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 points 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 for X values

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

00:19 This is one solution

00:28 This is the second solution

00:36 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

$y=(x-2)(x+4)$

With the X

Step-by-step solution

To solve for the points of intersection of the function $y = (x-2)(x+4)$ with the x-axis, we proceed as follows:

Set the function equal to zero to find the x-intercepts: $(x-2)(x+4) = 0$ .
Apply the zero-product property, which tells us that if a product of factors equals zero, then at least one of the factors must be zero. Thus, we solve the equations:
$x-2 = 0$ or $x+4 = 0$ .

Solving these equations, we find:

$x-2 = 0$ gives $x = 2$ .

$x+4 = 0$ gives $x = -4$ .

Therefore, the points of intersection with the x-axis are the points where $y=0$ . Substituting these x-values into $y = (x-2)(x+4)$ , we confirm that the corresponding y-values are zero:

For $x = 2$ , the point is $(2,0)$ .
For $x = -4$ , the point is $(-4,0)$ .

Thus, the points of intersection are $(-4,0)$ and $(2,0)$ .

Final Answer

$(-4,0),(2,0)$

Key Points to Remember

Essential concepts to master this topic

Zero Product Property: If a·b = 0, then a = 0 or b = 0
Technique: Set each factor equal to zero: (x-2) = 0 and (x+4) = 0
Check: X-intercepts have y-coordinate of 0: (-4,0) and (2,0) ✓

Common Mistakes

Avoid these frequent errors

Setting the function equal to y instead of zero
Don't solve y = (x-2)(x+4) for when y equals some value other than zero = wrong intersection points! X-intercepts occur only when the function crosses the x-axis at y = 0. Always set the entire function equal to zero to find x-intercepts.

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 I set the function equal to zero to find x-intercepts?

X-intercepts are points where the graph crosses the x-axis. At these points, the y-coordinate is always 0. So you solve $(x-2)(x+4) = 0$ to find where y = 0.

What if the quadratic wasn't already factored?

You'd need to factor it first or use other methods like the quadratic formula. The factored form $(x-2)(x+4)$ makes finding x-intercepts much easier!

How do I remember which coordinate is which?

Think alphabetically: x comes before y, so x-intercepts are written as (x-value, 0). The y-coordinate is always 0 for x-intercepts.

Can I expand the function first before solving?

You could expand to get $x^2 + 2x - 8 = 0$ and solve, but it's much faster to use the zero product property with the factored form!

What if one of my factors gives a decimal answer?

That's perfectly normal! Not all x-intercepts are whole numbers. Just solve each factor equation carefully and double-check your arithmetic.

How do I verify my x-intercepts are correct?

Substitute each x-value back into the original function. For x = 2: $(2-2)(2+4) = 0 \cdot 6 = 0$ ✓. For x = -4: $(-4-2)(-4+4) = (-6) \cdot 0 = 0$ ✓

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Solve the Quadratic Equation: Find Intersection Points of y=(x-2)(x+4)

X-Intercepts with Factored Quadratics

❤️ 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 I set the function equal to zero to find x-intercepts?

What if the quadratic wasn't already factored?

How do I remember which coordinate is which?

Can I expand the function first before solving?

What if one of my factors gives a decimal answer?

How do I verify my x-intercepts are correct?

🌟 Unlock Your Math Potential

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