Solve (x-5)² = 0: Finding X-Axis Intersections of a Quadratic Function

X-Axis Intersections with Perfect Square Equations

Find the intersection of the function

y=(x−5)2 y=(x-5)^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=0
00:08 Therefore, substitute Y=0 and solve to find the intersection point with the X-axis
00:12 Extract the root to eliminate the exponent
00:24 Isolate X
00:31 This is the X value at the intersection point, we substitute Y=0 as we did at the point
00:36 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Find the intersection of the function

y=(x−5)2 y=(x-5)^2

With the X

2

Step-by-step solution

To find the intersection of the parabola y=(x−5)2 y = (x-5)^2 with the x-axis, we must set the value of y y to zero since any point on the x-axis has a y-coordinate of zero.

Solving the equation:

  • Set (x−5)2=0 (x-5)^2 = 0 .
  • To solve for x x , remove the square by taking the square root of both sides. This yields x−5=0 x-5 = 0 .
  • Solve for x x by adding 5 to both sides: x=5 x = 5 .

Thus, the intersection point of the parabola with the x-axis is at (5,0) (5, 0) .

Therefore, the correct answer is (5,0) \boxed{(5,0)} , which corresponds to choice 3.

3

Final Answer

(5,0) (5,0)

Key Points to Remember

Essential concepts to master this topic
  • X-Axis Rule: Set y = 0 to find where parabolas cross horizontal axis
  • Technique: (x−5)2=0 (x-5)^2 = 0 means x−5=0 x-5 = 0 , so x=5 x = 5
  • Check: Substitute back: y=(5−5)2=02=0 y = (5-5)^2 = 0^2 = 0 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing x and y coordinates in the final answer
    Don't write (0,5) when x = 5 = wrong coordinate order! The x-axis intersection means y = 0, so the point is (x-value, 0). Always write x-axis intersections as (x, 0) where x is your solution.

Practice Quiz

Test your knowledge with interactive questions

Find the intersection of the function

\( y=(x+4)^2 \)

With the Y

FAQ

Everything you need to know about this question

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

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The x-axis is where y = 0 everywhere! Any point touching the x-axis has coordinates (something, 0). So to find where your parabola touches the x-axis, you need y = 0.

What does (x−5)2=0 (x-5)^2 = 0 actually mean?

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It means "what number, when you subtract 5 and square it, gives zero?" Since only zero squared equals zero, we need x - 5 = 0, which means x = 5.

Why is there only one intersection point instead of two?

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This parabola just touches the x-axis at one point (called the vertex). It doesn't cross through - it bounces off! Some parabolas cross twice, some once, some never.

How do I know which coordinate goes first?

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Always remember: (x, y) - x comes first! For x-axis intersections, y is always 0, so your answer looks like (some number, 0).

What if I got x = 25 instead?

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You probably expanded (x−5)2 (x-5)^2 unnecessarily! Don't expand - just take the square root: (x−5)2=x−5 \sqrt{(x-5)^2} = x-5 , then solve x−5=0 x-5 = 0 .

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