Quadratic Conundrum: Solving 2x² - 8 = x^2 + 4

Quadratic Equations with Square Root Solutions

Solve the following equation:


2x2−8=x2+4 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
00:29 Extract root
00:32 When extracting a root there are always 2 solutions (positive, negative)
00:35 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following equation:


2x2−8=x2+4 2x^2-8=x^2+4

2

Step-by-step solution

Let's solve the equation step-by-step:

  • Step 1: Rearrange the equation.

We start with the given equation:

2x2−8=x2+42x^2 - 8 = x^2 + 4

Subtract x2+4x^2 + 4 from both sides to get:

2x2−8−x2−4=02x^2 - 8 - x^2 - 4 = 0

  • Step 2: Simplify the equation.

Combine the like terms:

(2x2−x2)−8−4=0(2x^2 - x^2) - 8 - 4 = 0

This simplifies to:

x2−12=0x^2 - 12 = 0

  • Step 3: Solve for xx.

Add 12 to both sides:

x2=12x^2 = 12

Now take the square root of both sides:

x=±12x = \pm \sqrt{12}

Given the choices, the correct answer is ±12\pm \sqrt{12}.

3

Final Answer

±12 ±\sqrt{12}

Key Points to Remember

Essential concepts to master this topic
  • Rearrangement: Move all terms to one side first
  • Technique: Combine like terms: 2x2−x2=x2 2x^2 - x^2 = x^2
  • Check: Substitute x=±12 x = \pm\sqrt{12} back into original equation ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting the negative solution
    Don't write just x=12 x = \sqrt{12} = missing half the answer! When you take the square root of both sides, you get both positive and negative solutions because both (12)2=12 (\sqrt{12})^2 = 12 and (−12)2=12 (-\sqrt{12})^2 = 12 . Always include the ± symbol for complete solutions.

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 when solving x² = 12?

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Because squaring eliminates the sign! Both positive and negative numbers give the same result when squared. For example: 32=9 3^2 = 9 and (−3)2=9 (-3)^2 = 9 .

Should I simplify √12 further?

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You could! 12=4×3=23 \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} , so the answer could be written as ±23 \pm 2\sqrt{3} . However, both forms are correct.

How do I know when to move terms to one side?

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Always aim for the standard form ax2+bx+c=0 ax^2 + bx + c = 0 . Move all terms to the side that makes the leading coefficient positive for easier solving.

What if I combine the terms incorrectly?

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Take your time with like terms! Only combine terms with the same variable and power. Here: 2x2−x2=x2 2x^2 - x^2 = x^2 and −8−4=−12 -8 - 4 = -12 .

Can I solve this using the quadratic formula?

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Yes, but it's unnecessary! Since this simplifies to x2−12=0 x^2 - 12 = 0 , taking square roots is much faster than using x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} .

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