Find the Roots: Solving the Quadratic Equation x² - 16 = 0

Q: How do I know when to use difference of squares?

Look for the pattern $ a^2 - b^2 $! If you have two perfect squares separated by subtraction, like $ x^2 - 16 $, you can factor it as $ (a-b)(a+b) $.

Q: Why does this equation have two solutions?

Quadratic equations can have up to 2 solutions. When you factor $ (x-4)(x+4) = 0 $, each factor gives you one solution: x = 4 and x = -4.

Q: What if I can't recognize 16 as a perfect square?

Practice memorizing perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... Remember that $ 16 = 4^2 $, so $ x^2 - 16 = x^2 - 4^2 $.

Q: Can I solve this by taking the square root of both sides?

Yes! You can rewrite as $ x^2 = 16 $, then $ x = ±\sqrt{16} = ±4 $. This gives the same answers: x = 4 and x = -4.

Q: Why do both solutions work?

Both solutions satisfy the original equation! $ (-4)^2 - 16 = 16 - 16 = 0 $ ✓ and $ (4)^2 - 16 = 16 - 16 = 0 $ ✓. Always verify both answers!

Difference of Squares with Perfect Square Terms

Solve the following equation:

$x^2-16=0$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Isolate X

00:09 Extract root

00:14 When extracting a root there are always 2 solutions (positive, negative)

00:19 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:

$x^2-16=0$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Recognize the structure of the equation as a difference of squares.
Step 2: Factor the equation.
Step 3: Use the zero-product property to find the values of $x$ .

Now, let's work through each step:
Step 1: The equation given is $x^2 - 16 = 0$ . This equation is a type of difference of squares, as it can be expressed in the form $a^2 - b^2 = (a - b)(a + b)$ .
Step 2: Recognizing $16$ as $4^2$ , we can factor the equation: $x^2 - 16 = (x - 4)(x + 4) = 0$ .
Step 3: According to the zero-product property, if the product of two expressions is zero, then at least one of the expressions must be zero. Therefore, we set each factor equal to zero:
$x - 4 = 0$ or $x + 4 = 0$ .
Solving these simple linear equations gives $x = 4$ and $x = -4$ .

Therefore, the solutions to the equation are $x_1 = -4$ and $x_2 = 4$ .

Final Answer

$x_1=-4,x_2=4$

Key Points to Remember

Essential concepts to master this topic

Pattern: Recognize $x^2 - 16$ as difference of squares formula
Factoring: $x^2 - 16 = (x-4)(x+4) = 0$
Check: Substitute back: $(-4)^2 - 16 = 0$ and $4^2 - 16 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Setting equation equal to individual factors instead of zero
Don't solve x - 4 = x + 4 = you get no solution! This ignores that the original equation equals zero. Always set each factor equal to zero separately: x - 4 = 0 and x + 4 = 0.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$$ 2x^2-8=x^2+4 $$

FAQ

Everything you need to know about this question

How do I know when to use difference of squares?

Look for the pattern $a^2 - b^2$ ! If you have two perfect squares separated by subtraction, like $x^2 - 16$ , you can factor it as $(a-b)(a+b)$ .

Why does this equation have two solutions?

Quadratic equations can have up to 2 solutions. When you factor $(x-4)(x+4) = 0$ , each factor gives you one solution: x = 4 and x = -4.

What if I can't recognize 16 as a perfect square?

Practice memorizing perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... Remember that $16 = 4^2$ , so $x^2 - 16 = x^2 - 4^2$ .

Can I solve this by taking the square root of both sides?

Yes! You can rewrite as $x^2 = 16$ , then $x = ±\sqrt{16} = ±4$ . This gives the same answers: x = 4 and x = -4.

Why do both solutions work?

Both solutions satisfy the original equation! $(-4)^2 - 16 = 16 - 16 = 0$ ✓ and $(4)^2 - 16 = 16 - 16 = 0$ ✓. Always verify both answers!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Solving Quadratic Equations questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime