Solve for X in y=x²: Finding X-Coordinates When y=8

Q: Why do we get two answers for x?

Because both positive and negative numbers give positive results when squared! For example, $ (2\sqrt{2})^2 = 8 $ and $ (-2\sqrt{2})^2 = 8 $. The parabola y = x² crosses the line y = 8 at two points.

Q: How do I simplify √8?

Look for perfect square factors! Since 8 = 4 × 2, we get $ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} $. Always factor out perfect squares from under the radical.

Q: Can I leave my answer as ±√8?

While technically correct, it's better to simplify radicals when possible. $ ±2\sqrt{2} $ is the simplified form and shows you understand how to work with radicals.

Q: What if y was negative, like y = -8?

Then the equation $ x^2 = -8 $ would have no real solutions because you can't square a real number and get a negative result. The parabola y = x² never goes below the x-axis.

Quadratic Equations with Radical Solutions

What is the value of X for the function?

$y=x^2$

of the point $y=8$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Set up and solve

00:06 Substitute appropriate values according to the given data, and solve for X

00:23 Extract root

00:37 When extracting a root there are 2 solutions, positive and negative

00:42 Factor 8 into factors 4 and 2

00:48 Break down the root into 2 roots

00:52 Calculate root 4

00:56 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

What is the value of X for the function?

$y=x^2$

of the point $y=8$ ?

Step-by-step solution

The problem requires us to find the value of $x$ for the function $y = x^2$ when $y = 8$ .

Let's solve this step-by-step:

Step 1: Set up the equation based on the given function:
$\ y = x^2$ becomes $\ x^2 = 8$ .
Step 2: Solve for $x$ by taking the square root of both sides:

Since $\ x^2 = 8$ , we take the square root of both sides to find $x$ :

$x = \pm\sqrt{8}$

Step 3: Simplify the square root:

The square root of 8 can be simplified to $\ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$ .

Thus, $\ x = \pm 2\sqrt{2}$ .

Therefore, the solution is $x = \pm 2\sqrt{2}$ , which corresponds to choice 4.

Final Answer

$x=\pm2\sqrt{2}$

Key Points to Remember

Essential concepts to master this topic

Setup: Replace y with given value to create solvable equation
Technique: Take square root of both sides: $\sqrt{8} = 2\sqrt{2}$
Check: Verify both solutions: $(2\sqrt{2})^2 = 8$ and $(-2\sqrt{2})^2 = 8$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the ± when taking square roots
Don't write x = √8 without the ± symbol = missing half the solution! Every positive number has two square roots (positive and negative). Always include ± when solving x² = positive number.

Practice Quiz

Test your knowledge with interactive questions

Complete:

The missing value of the function point:

$$ f(x)=x^2 $$

$$ f(?)=16 $$

FAQ

Everything you need to know about this question

Why do we get two answers for x?

Because both positive and negative numbers give positive results when squared! For example, $(2\sqrt{2})^2 = 8$ and $(-2\sqrt{2})^2 = 8$ . The parabola y = x² crosses the line y = 8 at two points.

How do I simplify √8?

Look for perfect square factors! Since 8 = 4 × 2, we get $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$ . Always factor out perfect squares from under the radical.

Can I leave my answer as ±√8?

While technically correct, it's better to simplify radicals when possible. $±2\sqrt{2}$ is the simplified form and shows you understand how to work with radicals.

What if y was negative, like y = -8?

Then the equation $x^2 = -8$ would have no real solutions because you can't square a real number and get a negative result. The parabola y = x² never goes below the x-axis.

How do I check if my answer is right?

Substitute both values back into the original equation. For $x = 2\sqrt{2}$ : $y = (2\sqrt{2})^2 = 4 \times 2 = 8$ ✓. Same result for the negative value!

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