Solve for x: Understanding Absolute Values in |x²+4| = 40

Q: Why do we get two cases from one absolute value equation?

Because absolute value measures distance from zero! If $ |A| = 40 $, then A could be 40 or -40. That's why we solve both $ x^2 + 4 = 40 $ and $ x^2 + 4 = -40 $.

Q: What if x² + 4 = -40 has no solution?

That's completely normal! Since $ x^2 $ is always positive and we add 4, we get $ x^2 + 4 ≥ 4 $. It can never equal -40, so we discard this case and keep only the valid solutions.

Q: How do I know if x = ±6 is really correct?

Always substitute back! Try $ x = 6 $: $ |6^2 + 4| = |36 + 4| = |40| = 40 ✓ $Try $ x = -6 $: $ |(-6)^2 + 4| = |36 + 4| = |40| = 40 ✓ $

Q: Why don't we get x = ±4 as an answer?

Let's check: $ |4^2 + 4| = |16 + 4| = |20| = 20 ≠ 40 $. The answer $ x = ±4 $ doesn't satisfy our original equation, so it's incorrect.

Q: Can absolute value equations have no solutions?

Yes! If both cases lead to impossible equations (like $ x^2 = -5 $), then there are no real solutions. Always check each case carefully.

Absolute Value Equations with Quadratic Expressions

$|x^2+4|=40$

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Understand the problem

$|x^2+4|=40$

Step-by-step solution

To solve the equation $|x^2 + 4| = 40$ , we consider two cases based on the properties of absolute value:

Case 1: $x^2 + 4 = 40$
- Subtract 4 from both sides to isolate the quadratic term:
$x^2 = 40 - 4$
$x^2 = 36$
- Solving for $x$ , take the square root of both sides:
$x = \pm \sqrt{36}$
$x = \pm 6$
Case 2: $x^2 + 4 = -40$
- Subtract 4 from both sides:
$x^2 = -40 - 4$
$x^2 = -44$
- Since $x^2 = -44$ has no real number solutions (as the square of a real number cannot be negative), we discard this case.

Therefore, the only valid solutions are from Case 1: $x = \pm 6$ .

Thus, the solution to the equation $|x^2 + 4| = 40$ is $x = \pm 6$ .

Final Answer

$x=±6$

Key Points to Remember

Essential concepts to master this topic

Definition: $|A| = B$ means A = B or A = -B
Technique: Solve $x^2 + 4 = 40$ gives $x^2 = 36$ , so $x = ±6$
Check: Substitute $x = 6$ : $|36 + 4| = |40| = 40$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check the negative case
Don't solve only $x^2 + 4 = 40$ = missing valid solutions! Students often forget absolute value equations create two cases. Always check both $x^2 + 4 = 40$ and $x^2 + 4 = -40$ , even if one has no real solutions.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why do we get two cases from one absolute value equation?

Because absolute value measures distance from zero! If $|A| = 40$ , then A could be 40 or -40. That's why we solve both $x^2 + 4 = 40$ and $x^2 + 4 = -40$ .

What if x² + 4 = -40 has no solution?

That's completely normal! Since $x^2$ is always positive and we add 4, we get $x^2 + 4 ≥ 4$ . It can never equal -40, so we discard this case and keep only the valid solutions.

How do I know if x = ±6 is really correct?

Always substitute back! Try $x = 6$ : $|6^2 + 4| = |36 + 4| = |40| = 40 ✓$
Try $x = -6$ : $|(-6)^2 + 4| = |36 + 4| = |40| = 40 ✓$

Why don't we get x = ±4 as an answer?

Let's check: $|4^2 + 4| = |16 + 4| = |20| = 20 ≠ 40$ . The answer $x = ±4$ doesn't satisfy our original equation, so it's incorrect.

Can absolute value equations have no solutions?

Yes! If both cases lead to impossible equations (like $x^2 = -5$ ), then there are no real solutions. Always check each case carefully.

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