Solve y = x² - 49: Finding X-Intercepts of a Quadratic Function

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

The x-intercepts are points where the graph crosses the x-axis. On the x-axis, the y-coordinate is always 0. So we set $ y = 0 $ to find these crossing points!

Q: How do I recognize a difference of squares?

Look for the pattern $ a^2 - b^2 $. In our problem, $ x^2 - 49 = x^2 - 7^2 $ because 49 is a perfect square. This factors as $ (x-7)(x+7) $.

Q: What if I can't factor the expression?

If $ x^2 - 49 $ couldn't be factored, you could use the square root method: $ x^2 = 49 $, so $ x = ±\sqrt{49} = ±7 $. Both methods give the same answer!

Q: Why are there two x-intercepts?

Quadratic functions form parabolas, which typically cross the x-axis at two points (unless the vertex touches the x-axis). Since we got two solutions, x = 7 and x = -7, there are two x-intercepts.

Q: How do I write the final answer correctly?

X-intercepts are points, so write them as coordinates: $ (7, 0) $ and $ (-7, 0) $. Remember, the y-coordinate is always 0 for x-intercepts!

X-Intercepts with Difference of Squares

Determine the points of intersection of the function

$y=x^2-49$

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

00:12 Isolate X

00:18 Extract the root

00:25 Remember when extracting a root there are 2 solutions (positive and negative)

00:32 These are the X values

00:36 Y = 0 as we substituted at the beginning

00:42 And this is the solution to the problem

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

$y=x^2-49$

With the X

Step-by-step solution

To determine the points of intersection of the function $y = x^2 - 49$ with the x-axis, we set $y = 0$ . This gives us the equation:

x^2 - 49 = 0

We can solve this equation by factoring or using the square root method:

Recognize $x^2 - 49$ as a difference of squares since $49 = 7^2$ .
The equation can be rewritten and factored as: $(x - 7)(x + 7) = 0$ .

Setting each factor equal to zero gives:

$x - 7 = 0$ or $x + 7 = 0$

This simplifies to:

$x = 7$ or $x = -7$

Thus, the points of intersection are where the function crosses the x-axis, at the coordinates $(7, 0)$ and $(-7, 0)$ .

Referring to the given answer choices, the correct choice is:

$(7, 0), (-7, 0)$

Therefore, the points of intersection of the function $y = x^2 - 49$ with the x-axis are $(7, 0)$ and $(-7, 0)$ .

Final Answer

$(7,0),(-7,0)$

Key Points to Remember

Essential concepts to master this topic

X-Intercepts: Set y = 0 to find where function crosses x-axis
Factoring: Recognize $x^2 - 49 = (x-7)(x+7)$ as difference of squares
Check: Substitute x = 7 and x = -7 into original equation: both give y = 0 ✓

Common Mistakes

Avoid these frequent errors

Setting x = 0 instead of y = 0
Don't set x = 0 to find x-intercepts = gives you y-intercept (0, -49)! X-intercepts occur where the graph crosses the x-axis, meaning y = 0. Always set y = 0 to find x-intercepts.

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 we set y = 0 to find x-intercepts?

The x-intercepts are points where the graph crosses the x-axis. On the x-axis, the y-coordinate is always 0. So we set $y = 0$ to find these crossing points!

How do I recognize a difference of squares?

Look for the pattern $a^2 - b^2$ . In our problem, $x^2 - 49 = x^2 - 7^2$ because 49 is a perfect square. This factors as $(x-7)(x+7)$ .

What if I can't factor the expression?

If $x^2 - 49$ couldn't be factored, you could use the square root method: $x^2 = 49$ , so $x = ±\sqrt{49} = ±7$ . Both methods give the same answer!

Why are there two x-intercepts?

Quadratic functions form parabolas, which typically cross the x-axis at two points (unless the vertex touches the x-axis). Since we got two solutions, x = 7 and x = -7, there are two x-intercepts.

How do I write the final answer correctly?

X-intercepts are points, so write them as coordinates: $(7, 0)$ and $(-7, 0)$ . Remember, the y-coordinate is always 0 for x-intercepts!

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