Solve x² - 81 = 0: Finding Values in a Basic Quadratic Equation

Q: Why does this equation have two answers instead of one?

Quadratic equations usually have two solutions because when you square either a positive or negative number, you get the same result. For example: $ 9^2 = 81 $ and $ (-9)^2 = 81 $!

Q: How do I recognize the difference of squares pattern?

Look for the form $ a^2 - b^2 $. In this case, $ x^2 - 81 = x^2 - 9^2 $. The key is recognizing that 81 is a perfect square (9²).

Q: Can I solve this by adding 81 to both sides instead?

Yes! You could get $ x^2 = 81 $, then take the square root: $ x = ±\sqrt{81} = ±9 $. Both methods give the same answer, but factoring helps you see the structure better.

Q: What if I chose x = ±8 by mistake?

Check your work! Substitute back: $ 8^2 - 81 = 64 - 81 = -17 ≠ 0 $. This shows x = 8 is wrong. Always verify by substituting your answer back into the original equation.

Q: Why is the ± symbol so important here?

The ± symbol shows that both positive and negative values work. Writing just "x = 9" is incomplete because x = -9 is also a valid solution. Always include both when solving quadratics!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Isolate X

00:10 Extract the root

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

00:18 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve for x:

$x^2-81=0$

2

Step-by-step solution

Let's solve the given equation:

$x^2-81=0$ Note that we can factor the expression on the left side using the difference of squares formula:

$(\textcolor{red}{a}+\textcolor{blue}{b}) (\textcolor{red}{a}-\textcolor{blue}{b})=\textcolor{red}{a}^2-\textcolor{blue}{b}^2$ We'll do this using the fact that:

$81=9^2$ Therefore, we'll represent the rightmost term as a squared term:

$x^2-81=0 \\ \downarrow\\ \textcolor{red}{x}^2-\textcolor{blue}{9}^2=0$ If so, we can represent the expression on the left side in the above equation as a product of expressions:

$\textcolor{red}{x}^2-\textcolor{blue}{9}^2=0 \\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{9})(\textcolor{red}{x}-\textcolor{blue}{9})=0$ From here we'll remember that the product of expressions equals 0 only if at least one of the multiplied expressions equals zero,

Therefore, we'll get two simple equations and solve them by isolating the variable in each:

$x+9=0\\ \boxed{x=-9}$ or:

$x-9=0\\ \boxed{x=9}$

Let's summarize the solution of the equation:

$x^2-81=0 \\ \downarrow\\ \textcolor{red}{x}^2-\textcolor{blue}{9}^2=0 \\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{9})(\textcolor{red}{x}-\textcolor{blue}{9})=0 \\ x+9=0\rightarrow\boxed{x=-9}\\ x-9=0\rightarrow\boxed{x=9}\\ \downarrow\\ \boxed{x=9,-9}$ Therefore, the correct answer is answer B.

3

Final Answer

$x=\pm9$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Identify $x^2 - 81$ as difference of squares format
Factoring Technique: Rewrite as $(x+9)(x-9) = 0$ using $81 = 9^2$
Zero Product Property: Set each factor equal to zero: x = 9 or x = -9 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative solution
Don't solve only x - 9 = 0 and get just x = 9! This misses half the answer because quadratic equations typically have two solutions. Always solve both factors: x + 9 = 0 AND x - 9 = 0 to get x = ±9.

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 does this equation have two answers instead of one?

+

Quadratic equations usually have two solutions because when you square either a positive or negative number, you get the same result. For example: $9^2 = 81$ and $(-9)^2 = 81$ !

How do I recognize the difference of squares pattern?

+

Look for the form $a^2 - b^2$ . In this case, $x^2 - 81 = x^2 - 9^2$ . The key is recognizing that 81 is a perfect square (9²).

Can I solve this by adding 81 to both sides instead?

+

Yes! You could get $x^2 = 81$ , then take the square root: $x = ±\sqrt{81} = ±9$ . Both methods give the same answer, but factoring helps you see the structure better.

What if I chose x = ±8 by mistake?

+

Check your work! Substitute back: $8^2 - 81 = 64 - 81 = -17 ≠ 0$ . This shows x = 8 is wrong. Always verify by substituting your answer back into the original equation.

Why is the ± symbol so important here?

+

The ± symbol shows that both positive and negative values work. Writing just "x = 9" is incomplete because x = -9 is also a valid solution. Always include both when solving quadratics!

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Solve x² - 81 = 0: Finding Values in a Basic Quadratic Equation

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