Complete the Perfect Square: (□×x-□)² = 9x²-24x+16

Q: Why can't the answer be 4,3 instead of 3,4?

The order matters! The first blank is the coefficient of x, which comes from $ a^2 = 9 $, so $ a = 3 $. The second blank is the constant term, which comes from $ b^2 = 16 $, so $ b = 4 $.

Q: How do I know which square roots to choose (positive or negative)?

Start with positive values for simplicity! Since $ a^2 = 9 $ gives $ a = ±3 $ and $ b^2 = 16 $ gives $ b = ±4 $, check the middle term to determine signs. Here, $ -2ab = -24 $ works with $ a = 3, b = 4 $.

Q: What if I expand (3x - 4)² and don't get the right answer?

Double-check your expansion! $ (3x - 4)^2 = (3x)^2 - 2(3x)(4) + 4^2 = 9x^2 - 24x + 16 $. Make sure you're using the correct perfect square formula.

Q: Is there a faster way to solve this?

Yes! Notice that $ 9x^2 - 24x + 16 $ has coefficients 9, -24, 16. Since $ \sqrt{9} = 3 $ and $ \sqrt{16} = 4 $, check if $ 2(3)(4) = 24 $ matches the middle coefficient.

Q: Can perfect squares have negative leading coefficients?

Yes, but this problem gives us $ 9x^2 $ (positive), so we know $ a^2 = 9 $ is positive. The perfect square $ (ax - b)^2 $ always has a positive leading coefficient.

Perfect Square Trinomials with Coefficient Identification

Fill in the blanks:

$(?\times x-?)^2=9x^2-24x+16$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's fill in the missing part together.

00:13 Remember the shortcut for multiplying. We'll use these formulas to open up the brackets.

00:20 For the first term, find its square root. Easy, right?

00:25 Now, let's do the same for the last term.

00:32 Break down the middle term into factors: two, three, and four. You got this!

00:43 Next, let's check if the multiplication is correct.

00:48 Substitute each number carefully in its right place.

00:52 Great job! That's how we find the solution to the question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the blanks:

$(?\times x-?)^2=9x^2-24x+16$

Step-by-step solution

To solve this problem, let's identify the steps needed to determine the missing values.

Step 1: Apply the perfect square formula.
Write $(a \times x - b)^2 = a^2x^2 - 2abx + b^2$ .
Step 2: Compare with the given polynomial.
Set the expression equal to the provided polynomial: $9x^2 - 24x + 16$ .
Step 3: Match coefficients.
From $a^2x^2 = 9x^2$ , we get $a^2 = 9$ .
From $-2abx = -24x$ , we get $-2ab = -24$ .
From $b^2 = 16$ , we get $b^2 = 16$ .
Step 4: Solve for $a$ and $b$ .
- Solve $a^2 = 9$ , giving $a = 3$ or $a = -3$ . Choose $a = 3$ for simplicity.
- Solve $b^2 = 16$ , giving $b = 4$ or $b = -4$ . Choose $b = 4$ for simplicity.
- Check $-2ab = -24$ : $-2(3)(4) = -24$ , which is correct.

Therefore, the missing values are $\boxed{3, 4}$ .

Thus, the solution to the problem is: $3, 4$ .

Final Answer

$3,\text{ }4$

Key Points to Remember

Essential concepts to master this topic

Formula: Perfect square $(ax - b)^2 = a^2x^2 - 2abx + b^2$
Technique: Match coefficients: $a^2 = 9$ gives $a = 3$ , $b^2 = 16$ gives $b = 4$
Check: Verify middle term: $-2ab = -2(3)(4) = -24$ matches -24x ✓

Common Mistakes

Avoid these frequent errors

Guessing values without using the perfect square formula
Don't just try random numbers like 4,3 or 5,2 = wrong answers! This ignores the systematic relationship between coefficients. Always use the perfect square formula $(ax - b)^2 = a^2x^2 - 2abx + b^2$ and match each coefficient.

Practice Quiz

Test your knowledge with interactive questions

$$ (4b-3)(4b-3) $$

Rewrite the above expression as an exponential summation expression:

FAQ

Everything you need to know about this question

Why can't the answer be 4,3 instead of 3,4?

The order matters! The first blank is the coefficient of x, which comes from $a^2 = 9$ , so $a = 3$ . The second blank is the constant term, which comes from $b^2 = 16$ , so $b = 4$ .

How do I know which square roots to choose (positive or negative)?

Start with positive values for simplicity! Since $a^2 = 9$ gives $a = ±3$ and $b^2 = 16$ gives $b = ±4$ , check the middle term to determine signs. Here, $-2ab = -24$ works with $a = 3, b = 4$ .

What if I expand (3x - 4)² and don't get the right answer?

Double-check your expansion! $(3x - 4)^2 = (3x)^2 - 2(3x)(4) + 4^2 = 9x^2 - 24x + 16$ . Make sure you're using the correct perfect square formula.

Is there a faster way to solve this?

Yes! Notice that $9x^2 - 24x + 16$ has coefficients 9, -24, 16. Since $\sqrt{9} = 3$ and $\sqrt{16} = 4$ , check if $2(3)(4) = 24$ matches the middle coefficient.

Can perfect squares have negative leading coefficients?

Yes, but this problem gives us $9x^2$ (positive), so we know $a^2 = 9$ is positive. The perfect square $(ax - b)^2$ always has a positive leading coefficient.

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