Solve the Quadratic Equation: 2x² - 8 = x² + 4

Q: Why do I get two answers for this quadratic equation?

Quadratic equations typically have two solutions because when you square a number, both positive and negative values give the same result. For example, both $ (\sqrt{12})^2 = 12 $ and $ (-\sqrt{12})^2 = 12 $.

Q: Can I simplify √12 further?

Yes! $ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} $. So your final answer could be written as $ x = \pm 2\sqrt{3} $.

Q: What's the difference of squares pattern?

The pattern is $ a^2 - b^2 = (a-b)(a+b) $. Here, $ x^2 - 12 $ becomes $ x^2 - (\sqrt{12})^2 $, so a = x and b = √12.

Q: Do I always need to move everything to one side?

For quadratic equations, yes! Moving all terms to one side gives you the standard form $ ax^2 + bx + c = 0 $, which makes it easier to factor or use other solving methods.

Q: How do I check if my answers are correct?

Substitute each solution back into the original equation $ 2x^2 - 8 = x^2 + 4 $. Both $ x = \sqrt{12} $ and $ x = -\sqrt{12} $ should make both sides equal.

Quadratic Equations with Difference of Squares

Solve the following exercise:

$2x^2-8=x^2+4$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Isolate X

00:34 Extract root

00:38 When extracting a root there are always 2 solutions (positive, negative)

00:41 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

Solve the following exercise:

$2x^2-8=x^2+4$

Step-by-step solution

First, we move the terms to one side equal to 0.

$2x^2-x^2-8-4=0$

We simplify :

$x^2-12=0$

We use the shortcut multiplication formula to solve:

$x^2-(\sqrt{12})^2=0$

$(x-\sqrt{12})(x+\sqrt{12})=0$

$x=\pm\sqrt{12}$

Final Answer

$±\sqrt{12}$

Key Points to Remember

Essential concepts to master this topic

Rule: Move all terms to one side to create standard form
Technique: Recognize $x^2 - 12 = (x - \sqrt{12})(x + \sqrt{12})$
Check: Substitute both solutions: $(\pm\sqrt{12})^2 - 12 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the ± symbol in the final answer
Don't write just $x = \sqrt{12}$ = incomplete solution! When you factor $x^2 - 12 = 0$ , both factors equal zero, giving two solutions. Always include ± to show both $+\sqrt{12}$ and $-\sqrt{12}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ 2x^2-8=x^2+4 $$

FAQ

Everything you need to know about this question

Why do I get two answers for this quadratic equation?

Quadratic equations typically have two solutions because when you square a number, both positive and negative values give the same result. For example, both $(\sqrt{12})^2 = 12$ and $(-\sqrt{12})^2 = 12$ .

Can I simplify √12 further?

Yes! $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ . So your final answer could be written as $x = \pm 2\sqrt{3}$ .

What's the difference of squares pattern?

The pattern is $a^2 - b^2 = (a-b)(a+b)$ . Here, $x^2 - 12$ becomes $x^2 - (\sqrt{12})^2$ , so a = x and b = √12.

Do I always need to move everything to one side?

For quadratic equations, yes! Moving all terms to one side gives you the standard form $ax^2 + bx + c = 0$ , which makes it easier to factor or use other solving methods.

How do I check if my answers are correct?

Substitute each solution back into the original equation $2x^2 - 8 = x^2 + 4$ . Both $x = \sqrt{12}$ and $x = -\sqrt{12}$ should make both sides equal.

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