Solve the Absolute Value Quadratic: |x²-11|=5

Q: Why does an absolute value equation have multiple solutions?

The absolute value $ |x^2-11| = 5 $ means the expression inside could equal 5 or -5. Each case gives you a quadratic equation with 2 solutions, so you get 4 total solutions!

Q: How do I know which case to use first?

It doesn't matter! You can solve $ x^2 - 11 = 5 $ or $ x^2 - 11 = -5 $ first. Just make sure to solve both cases completely.

Q: What if one of my cases has no real solutions?

That's okay! If a case gives you something like $ x^2 = -3 $, there are no real solutions from that case. Just use the solutions from the valid case.

Q: How can I check if all my answers are correct?

Substitute each solution back into the original equation $ |x^2-11| = 5 $. For example: if x = 4, then $ |16-11| = |5| = 5 $ ✓

Q: Why does the answer say 'Answers a + b'?

This means you need both sets of solutions: the solutions from option (a) which are $ x = ±4 $ AND the solutions from option (b) which are $ x = ±\sqrt{6} $.

Absolute Value Equations with Two Cases

$|x^2-11|=5$

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Understand the problem

$|x^2-11|=5$

Step-by-step solution

To solve the equation $|x^2 - 11| = 5$ , we will consider the two cases implied by the property of absolute values:

Case 1: $x^2 - 11 = 5$
Case 2: $x^2 - 11 = -5$

Let's solve each case separately.

Case 1: $x^2 - 11 = 5$
First, add 11 to both sides: $x^2 = 16$ .
Taking the square root of both sides, we get $x = \pm\sqrt{16} = \pm4$ .
Thus, the solutions for Case 1 are $x = 4$ and $x = -4$ .

Case 2: $x^2 - 11 = -5$
First, add 11 to both sides: $x^2 = 6$ .
Taking the square root of both sides, we get $x = \pm\sqrt{6}$ .
Thus, the solutions for Case 2 are $x = \sqrt{6}$ and $x = -\sqrt{6}$ .

Therefore, combining solutions from both cases, the complete set of solutions is $x = 4, -4, \sqrt{6}, -\sqrt{6}$ .

Reviewing the multiple-choice options, both cases' solutions correspond to the correct answer listed as 'Answers a + b.'

Final Answer

Answers a + b

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equations create two separate cases to solve
Technique: Solve $x^2 - 11 = 5$ and $x^2 - 11 = -5$ separately
Check: Test all solutions: $|16-11| = 5$ and $|6-11| = 5$ ✓

Common Mistakes

Avoid these frequent errors

Solving only one case of the absolute value
Don't solve just $x^2 - 11 = 5$ = only 2 solutions instead of 4! This misses half the correct answers because absolute value creates both positive and negative possibilities. Always solve both $x^2 - 11 = 5$ AND $x^2 - 11 = -5$ .

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why does an absolute value equation have multiple solutions?

The absolute value $|x^2-11| = 5$ means the expression inside could equal 5 or -5. Each case gives you a quadratic equation with 2 solutions, so you get 4 total solutions!

How do I know which case to use first?

It doesn't matter! You can solve $x^2 - 11 = 5$ or $x^2 - 11 = -5$ first. Just make sure to solve both cases completely.

What if one of my cases has no real solutions?

That's okay! If a case gives you something like $x^2 = -3$ , there are no real solutions from that case. Just use the solutions from the valid case.

How can I check if all my answers are correct?

Substitute each solution back into the original equation $|x^2-11| = 5$ . For example: if x = 4, then $|16-11| = |5| = 5$ ✓

Why does the answer say 'Answers a + b'?

This means you need both sets of solutions: the solutions from option (a) which are $x = ±4$ AND the solutions from option (b) which are $x = ±\sqrt{6}$ .

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