Solve for X in the Quadratic Equation: x² + 16x + 64 = 81

Q: How do I know if something is a perfect square trinomial?

Look for the pattern $ a^2 + 2ab + b^2 $. In $ x^2 + 16x + 64 $, we have x² (first square), 16x = 2(x)(8) (double product), and 64 = 8² (second square).

Q: What if I can't see the perfect square pattern?

You can always use the quadratic formula or try completing the square! Move 81 to the left side first: $ x^2 + 16x - 17 = 0 $, then apply your preferred method.

Q: Why do I get two answers from one equation?

Quadratic equations naturally have two solutions because when you square a number, both positive and negative values give the same result. For example, both 9² = 81 and (-9)² = 81.

Q: How can I check both answers quickly?

For x = 1: $ 1 + 16 + 64 = 81 $ ✓For x = -17: $ 289 + (-272) + 64 = 81 $ ✓Both work!

Quadratic Equations with Perfect Square Factoring

$x^2+16x+64=81$

Find X

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Factor 64 into 8 squared

00:11 Factor 16 into 2 and 8

00:20 Use the shortened multiplication formulas to find the brackets

00:25 Extract the root and find the two possible solutions

00:37 Isolate X

00:41 This is one solution

00:47 And this is the second solution

00:50 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

$x^2+16x+64=81$

Find X

Step-by-step solution

Let's solve the equation $x^2 + 16x + 64 = 81$ step-by-step:

Step 1: Recognize the Left-Hand Side as a Perfect Square
Notice that the left-hand side, $x^2 + 16x + 64$ , can be factored as $(x + 8)^2$ because $(x + 8)^2 = x^2 + 16x + 64$ .
Step 2: Set the Factored Expression Equal to 81
We have $(x + 8)^2 = 81$ .
Step 3: Take the Square Root of Both Sides
Taking the square root on both sides gives two possible equations: $x + 8 = 9$ and $x + 8 = -9$ .
Step 4: Solve Each Equation
For $x + 8 = 9$ :
Subtract 8 from both sides to get $x = 1$ .
For $x + 8 = -9$ :
Subtract 8 from both sides to get $x = -17$ .

Therefore, the solutions to the equation are $x = -17$ or $x = 1$ .

Final Answer

$x=-17$ or $x=1$

Key Points to Remember

Essential concepts to master this topic

Recognition: Identify perfect square trinomials like $x^2 + 16x + 64$
Technique: Factor as $(x + 8)^2 = 81$ then take square roots
Check: Substitute x = 1: $1 + 16 + 64 = 81$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative square root solution
Don't solve $(x + 8)^2 = 81$ as only x + 8 = 9, giving x = 1 = incomplete answer! Taking square roots produces both positive AND negative values. Always write x + 8 = ±9 to find both solutions x = 1 and x = -17.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+y)^2 $$

FAQ

Everything you need to know about this question

How do I know if something is a perfect square trinomial?

Look for the pattern $a^2 + 2ab + b^2$ . In $x^2 + 16x + 64$ , we have x² (first square), 16x = 2(x)(8) (double product), and 64 = 8² (second square).

What if I can't see the perfect square pattern?

You can always use the quadratic formula or try completing the square! Move 81 to the left side first: $x^2 + 16x - 17 = 0$ , then apply your preferred method.

Why do I get two answers from one equation?

Quadratic equations naturally have two solutions because when you square a number, both positive and negative values give the same result. For example, both 9² = 81 and (-9)² = 81.

Do I always need to rearrange the equation first?

Not always! In this problem, the left side was already a perfect square. But if you see something like $x^2 + 16x = 17$ , then yes - add 64 to complete the square.

How can I check both answers quickly?

For x = 1: $1 + 16 + 64 = 81$ ✓
For x = -17: $289 + (-272) + 64 = 81$ ✓

Both work!

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