Solving |x²+4| = 20: An Absolute Value Equation Challenge

Q: Why do I need to solve two separate equations?

The absolute value symbol means "distance from zero," so $ |A|=20 $ means A could be +20 or -20. That's why we solve both $ x^2+4=20 $ and $ x^2+4=-20 $!

Q: What if one of the equations gives no real solutions?

That's normal! In this problem, $ x^2=-24 $ has no real solutions because squares can't be negative. Just use the solutions from the equation that does work.

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

It doesn't matter which case you solve first! Always solve both cases and collect all the real solutions. The order won't change your final answer.

Q: Can the expression inside the absolute value be negative?

Yes! The expression $ x^2+4 $ is always positive (since $ x^2≥0 $ and we add 4), but in general, absolute value expressions can be negative before taking the absolute value.

Q: Why do we get ± when taking square roots?

When we solve $ x^2=16 $, both positive and negative numbers can be squared to get 16. Since $ (+4)^2=16 $ and $ (-4)^2=16 $, we write $ x=±4 $.

Absolute Value Equations with Quadratic Expressions

$|x^2+4|=20$

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Understand the problem

$|x^2+4|=20$

Step-by-step solution

To solve the equation $|x^2 + 4| = 20$ , we need to consider the definition of absolute value, which gives us two equations to solve:

First equation: $x^2 + 4 = 20$
Second equation: $x^2 + 4 = -20$

Let's solve each one individually:

For the first equation, $x^2 + 4 = 20$ :

Subtract 4 from both sides to isolate the $x^2$ term:

$x^2 = 20 - 4$

$x^2 = 16$

Taking the square root of both sides gives us the solutions:

$x = \pm \sqrt{16}$

$x = \pm 4$

Now, consider the second equation, $x^2 + 4 = -20$ :

Subtract 4 from both sides:

$x^2 = -20 - 4$

$x^2 = -24$

This provides no real solutions since a square cannot be negative in the real number system.

Therefore, the solutions to the original equation are $x = \pm 4$ , which corresponds to choice 2.

Final Answer

$x=±4$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equation creates two separate cases to solve
Technique: For $|x^2+4|=20$ , solve $x^2+4=20$ and $x^2+4=-20$
Check: Substitute $x=4$ : $|16+4|=|20|=20$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to consider both positive and negative cases
Don't just solve $x^2+4=20$ and stop = missing half the solutions! The absolute value means the expression inside could equal +20 or -20. Always set up both equations: positive case AND negative case.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why do I need to solve two separate equations?

The absolute value symbol means "distance from zero," so $|A|=20$ means A could be +20 or -20. That's why we solve both $x^2+4=20$ and $x^2+4=-20$ !

What if one of the equations gives no real solutions?

That's normal! In this problem, $x^2=-24$ has no real solutions because squares can't be negative. Just use the solutions from the equation that does work.

How do I know which case to use first?

It doesn't matter which case you solve first! Always solve both cases and collect all the real solutions. The order won't change your final answer.

Can the expression inside the absolute value be negative?

Yes! The expression $x^2+4$ is always positive (since $x^2≥0$ and we add 4), but in general, absolute value expressions can be negative before taking the absolute value.

Why do we get ± when taking square roots?

When we solve $x^2=16$ , both positive and negative numbers can be squared to get 16. Since $(+4)^2=16$ and $(-4)^2=16$ , we write $x=±4$ .

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