Solve the Quadratic Equation: x²-8x+16=0 Using Perfect Square Method

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

Check if the middle term equals twice the product of the square roots of the first and last terms. For $ x^2 - 8x + 16 $: $ \sqrt{x^2} = x $, $ \sqrt{16} = 4 $, so $ 2 \cdot x \cdot 4 = 8x $ ✓

Q: Why does (x-4)² = 0 give only one solution?

When you take the square root of both sides, you get $ x - 4 = 0 $ because $ \sqrt{0} = 0 $. This is called a repeated root - the same solution appears twice!

Q: Can I use the quadratic formula instead?

Yes! The quadratic formula will give $ x = \frac{8 \pm \sqrt{64-64}}{2} = \frac{8}{2} = 4 $. But recognizing perfect squares is faster and shows the structure of the equation better.

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

Look for clues: the first and last terms should be perfect squares, and check if $ b^2 = 4ac $ (discriminant equals zero). Practice recognizing common patterns like $ x^2 \pm 6x + 9 $!

Q: Do all quadratic equations have perfect square forms?

No! Only special quadratics factor as perfect squares. Most require other methods like factoring $ (x+m)(x+n) $ or the quadratic formula.

Perfect Square Trinomials with Repeated Roots

$x^2-8x+16=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find the value of X.

00:09 First, change 16 to 4 squared.

00:20 Next, break it down into smaller parts.

00:26 Now, use quick multiplication tricks to get the product.

00:31 Look for numbers that make each section equal zero.

00:36 And that's how we solve this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^2-8x+16=0$

Step-by-step solution

Let's solve the given equation:

$x^2-8x+16=0$ We identify that we can factor the expression on the left side using the perfect square trinomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$ Let's do this:

$x^2-8x+16=0 \\ x^2\textcolor{blue}{-8x}+4^2=0 \\ x^2\textcolor{blue}{-2\cdot x\cdot4}+4^2=0 \\ \downarrow\\ (x-4)^2=0$ Note that factoring using this formula was only possible because the middle term in the expression (which is in first power in this case and highlighted in blue in the previous calculation) indeed matched the middle term in the perfect square trinomial formula,

We'll continue and solve the resulting equation by taking the square root of both sides:

$(x-4)^2=0 \hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\\ x-4=0\\ \boxed{x=4}$ Therefore, the correct answer is answer C.

Final Answer

$x=4$

Key Points to Remember

Essential concepts to master this topic

Pattern: Recognize $a^2 \pm 2ab + b^2 = (a \pm b)^2$ form
Technique: Factor $x^2 - 8x + 16 = (x-4)^2$ using perfect square
Check: Substitute x=4: $4^2 - 8(4) + 16 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check if trinomial is a perfect square
Don't assume every quadratic can be factored as a perfect square = wrong factoring! You must verify the middle term matches 2ab pattern. Always check: does -8x equal -2·x·4? Only then can you write (x-4)².

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

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

Check if the middle term equals twice the product of the square roots of the first and last terms. For $x^2 - 8x + 16$ : $\sqrt{x^2} = x$ , $\sqrt{16} = 4$ , so $2 \cdot x \cdot 4 = 8x$ ✓

Why does (x-4)² = 0 give only one solution?

When you take the square root of both sides, you get $x - 4 = 0$ because $\sqrt{0} = 0$ . This is called a repeated root - the same solution appears twice!

Can I use the quadratic formula instead?

Yes! The quadratic formula will give $x = \frac{8 \pm \sqrt{64-64}}{2} = \frac{8}{2} = 4$ . But recognizing perfect squares is faster and shows the structure of the equation better.

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

Look for clues: the first and last terms should be perfect squares, and check if $b^2 = 4ac$ (discriminant equals zero). Practice recognizing common patterns like $x^2 \pm 6x + 9$ !

Do all quadratic equations have perfect square forms?

No! Only special quadratics factor as perfect squares. Most require other methods like factoring $(x+m)(x+n)$ or the quadratic formula.

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