Factor the Quadratic Polynomial: Simplifying x² - 16x + 64

Q: Why is the answer (x-8)² and not (x+8)²?

The middle term sign determines this! Since we have $ -16x $ (negative), we need $ (x-8)^2 $. If it were $ +16x $, then we'd use $ (x+8)^2 $.

Q: Can I use the quadratic formula instead of factoring?

Yes, but factoring is much faster for perfect squares! The quadratic formula will give you $ x = 8 $ (repeated root), but recognizing the pattern saves time and shows the structure clearly.

Q: What if the first coefficient isn't 1?

Factor out the coefficient first! For example, $ 4x^2 - 32x + 64 = 4(x^2 - 8x + 16) = 4(x-4)^2 $. Always look for common factors before applying perfect square patterns.

Q: How do I check my factoring is correct?

Expand your factored form using FOIL or the square pattern. $ (x-8)^2 = (x-8)(x-8) = x^2 - 8x - 8x + 64 = x^2 - 16x + 64 $ ✓

Perfect Square Trinomials with Factoring Methods

Find the representation of the product of the following function

$f(x)=x^2-16x+64$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Convert metrinum to multiplication

00:05 Break down 64 into 8 squared

00:11 Break down 16 into factors 2 and 8

00:19 Using these factors, we'll create the appropriate multiplication

00:24 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

Find the representation of the product of the following function

$f(x)=x^2-16x+64$

Step-by-step solution

To find the product representation of the function $f(x) = x^2 - 16x + 64$ , we expect it to be a perfect square trinomial.

First, recognize that the given quadratic form is $a^2 - 2ab + b^2$ . Comparing it with $x^2 - 16x + 64$ :

The first term $x^2$ suggests $a = x$ .
The middle term $-16x$ can be interpreted as $-2ab$ . So, $-2a = -16$ , solving gives $2b = 16$ , hence $b = 8$ .
The last term $(64)$ is $b^2$ , confirms that $8^2 = 64$ .

This means our expression is:

$f(x) = (x - 8)^2$

Thus, the product form or factored representation of the function is $(x-8)^2$ .

The final answer is: $(x-8)^2$ .

Final Answer

$(x-8)^2$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Perfect square trinomials follow $a^2 - 2ab + b^2$ format
Middle Term Check: Verify -16x equals -2(x)(8) = -16x exactly
Verification: Expand $(x-8)^2$ to get $x^2 - 16x + 64$ ✓

Common Mistakes

Avoid these frequent errors

Confusing sign patterns in perfect square trinomials
Don't factor $x^2 - 16x + 64$ as $(x+8)^2$ = $x^2 + 16x + 64$ ! The middle term becomes positive, not negative. Always match the middle term sign: negative coefficient means $(x-8)^2$ .

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

$$ a=2,b=2,c=2 $$

FAQ

Everything you need to know about this question

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

Check if the first and last terms are perfect squares, then verify the middle term equals twice the product of their square roots. For $x^2 - 16x + 64$ : $\sqrt{x^2} = x$ , $\sqrt{64} = 8$ , and $2(x)(8) = 16x$ matches!

Why is the answer (x-8)² and not (x+8)²?

The middle term sign determines this! Since we have $-16x$ (negative), we need $(x-8)^2$ . If it were $+16x$ , then we'd use $(x+8)^2$ .

Can I use the quadratic formula instead of factoring?

Yes, but factoring is much faster for perfect squares! The quadratic formula will give you $x = 8$ (repeated root), but recognizing the pattern saves time and shows the structure clearly.

What if the first coefficient isn't 1?

Factor out the coefficient first! For example, $4x^2 - 32x + 64 = 4(x^2 - 8x + 16) = 4(x-4)^2$ . Always look for common factors before applying perfect square patterns.

How do I check my factoring is correct?

Expand your factored form using FOIL or the square pattern. $(x-8)^2 = (x-8)(x-8) = x^2 - 8x - 8x + 64 = x^2 - 16x + 64$ ✓

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Factor the Quadratic Polynomial: Simplifying x² - 16x + 64

Perfect Square Trinomials with Factoring Methods

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

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

Why is the answer (x-8)² and not (x+8)²?

Can I use the quadratic formula instead of factoring?

What if the first coefficient isn't 1?

How do I check my factoring is correct?

🌟 Unlock Your Math Potential

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