Solve the Quadratic Equation: x² - 16 = x + 4 Step-by-Step

Q: Why can't I just divide by (x+4) like the solution shows?

Be very careful! Dividing by (x+4) assumes x ≠ -4. While it works here, it's safer to use standard methods: rearrange to $ x^2 - x - 20 = 0 $, then factor or use the quadratic formula.

Q: Is there another way to solve this equation?

Yes! Rearrange to $ x^2 - x - 20 = 0 $, then factor as $ (x-5)(x+4) = 0 $. This gives x = 5 or x = -4. Check both solutions in the original equation.

Q: How do I know if x = -4 is also a solution?

Substitute x = -4 into the original equation: $ (-4)^2 - 16 = -4 + 4 $ gives $ 16 - 16 = 0 $, so $ 0 = 0 $ ✓. So x = -4 is actually also a valid solution!

Q: Why does the given answer only show x = 5?

The method shown accidentally eliminated the solution x = -4 by dividing by (x+4). The complete solution set should be x = 5 or x = -4. Always verify all potential solutions!

Q: What's the difference of squares pattern?

The pattern is $ a^2 - b^2 = (a-b)(a+b) $. Here, $ x^2 - 16 = x^2 - 4^2 = (x-4)(x+4) $. This makes factoring much faster!

Quadratic Equations with Factoring Division Method

Solve the following equation:

$x^2-16=x+4$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Use the shortened multiplication formulas

00:15 Break down 16 into 4 squared

00:21 Use the shortened multiplication formulas and build a trinomial

00:28 Divide by the common factor

00:41 Reduce what we can

00:47 Isolate X

00:55 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 equation:

$x^2-16=x+4$

Step-by-step solution

Please note that the equation can be arranged differently:

x²-16 = x +4

x² - 4² = x +4

We will first factor a trinomial for the section on the left

(x-4)(x+4) = x+4

We will then divide everything by x+4

(x-4)(x+4) / x+4 = x+4 / x+4

x-4 = 1

x = 5

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rearrangement: Move all terms to one side before factoring
Factoring: Recognize $x^2 - 16 = (x-4)(x+4)$ as difference of squares
Check: Substitute x = 5: $25 - 16 = 5 + 4$ , so 9 = 9 ✓

Common Mistakes

Avoid these frequent errors

Dividing by a variable expression without checking restrictions
Don't divide both sides by (x+4) without considering x ≠ -4! This could eliminate valid solutions or create false ones. Always check that your division doesn't make the denominator zero, and use standard quadratic methods instead.

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 can't I just divide by (x+4) like the solution shows?

Be very careful! Dividing by (x+4) assumes x ≠ -4. While it works here, it's safer to use standard methods: rearrange to $x^2 - x - 20 = 0$ , then factor or use the quadratic formula.

Is there another way to solve this equation?

Yes! Rearrange to $x^2 - x - 20 = 0$ , then factor as $(x-5)(x+4) = 0$ . This gives x = 5 or x = -4. Check both solutions in the original equation.

How do I know if x = -4 is also a solution?

Substitute x = -4 into the original equation: $(-4)^2 - 16 = -4 + 4$ gives $16 - 16 = 0$ , so $0 = 0$ ✓. So x = -4 is actually also a valid solution!

Why does the given answer only show x = 5?

The method shown accidentally eliminated the solution x = -4 by dividing by (x+4). The complete solution set should be x = 5 or x = -4. Always verify all potential solutions!

What's the difference of squares pattern?

The pattern is $a^2 - b^2 = (a-b)(a+b)$ . Here, $x^2 - 16 = x^2 - 4^2 = (x-4)(x+4)$ . This makes factoring much faster!

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