Complete the Perfect Square: Solve (x-?)² = x²-?+25

Q: Why is the first blank 5 and not -5?

Great question! In $ (x-?)^2 $, the question mark represents the value being subtracted, not a negative number. Since $ a^2 = 25 $, we have $ a = 5 $, making it $ (x-5)^2 $.

Q: How do I find the middle term?

Use the perfect square formula! For $ (x-a)^2 $, the middle term is always $ -2ax $. Since $ a = 5 $, the middle term is $ -2(5)x = -10x $. But the blank asks for the positive coefficient, so it's 10x.

Q: Can I check my answer by expanding?

Absolutely! That's the best way to verify. Expand $ (x-5)^2 $: $ (x-5)(x-5) = x^2 - 5x - 5x + 25 = x^2 - 10x + 25 $. This matches the given form perfectly!

Q: Why does the problem show the middle term as positive 10x in the answer?

The equation format is $ x^2 - ? + 25 $, where the blank represents the coefficient of the middle term. Since the actual middle term is $ -10x $, the coefficient is 10x (the minus sign is already shown).

Perfect Square Trinomials with Missing Terms

Fill in the blanks:

$(x-?)^2=x^2-?+25$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing

00:03 Use the shortened multiplication formulas to open the parentheses

00:06 Let's substitute A as the unknown

00:14 Let's equate the equal coefficients

00:18 Let's extract the root and find A

00:21 Let's substitute A and solve

00:34 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

Fill in the blanks:

$(x-?)^2=x^2-?+25$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Recognize this as a binomial square problem.
Step 2: Identify the expanded form of a binomial square.
Step 3: Match the terms from both sides of the equation.

Now, let's work through each step:
Step 1: The given expression $(x-?)^2 = x^2 - ? + 25$ is the expansion of a binomial $(x-a)^2$ .
Step 2: Recall that $(x-a)^2 = x^2 - 2ax + a^2$ .
Step 3: Compare this equivalent form to $x^2 - ? + 25$ :

- The term $x^2$ matches directly on both sides.
- The constant term $a^2 = 25$ , so $a = 5$ because $5^2 = 25$ .
- The middle term $-2ax$ is currently unspecified, but it provides the form needed to fill the blank with $-2ax$ .

Matching the expanded form: - The term inside $(x-?)$ should be $5$ (because $a=5$ ).
- Therefore, the missing linear term can be $10x$ since $-2 \times 5 = -10$ .

Therefore, the solution to the problem is $5,\text{ }10x$ .

Final Answer

$5,\text{ }10x$

Key Points to Remember

Essential concepts to master this topic

Formula: $(x-a)^2 = x^2 - 2ax + a^2$ for any value a
Technique: From constant term 25, find $a = \sqrt{25} = 5$ , then middle term is $-2(5)x = -10x$
Check: Expand $(x-5)^2 = x^2 - 10x + 25$ matches the given form ✓

Common Mistakes

Avoid these frequent errors

Taking the negative square root for the constant term
Don't assume the first blank is -5 just because you see a minus sign = wrong expansion! The minus sign is part of the binomial form (x-a)², not the value of a itself. Always take the positive square root of the constant term to find a.

Practice Quiz

Test your knowledge with interactive questions

Declares the given expression as a sum

$$ (7b-3x)^2 $$

FAQ

Everything you need to know about this question

Why is the first blank 5 and not -5?

Great question! In $(x-?)^2$ , the question mark represents the value being subtracted, not a negative number. Since $a^2 = 25$ , we have $a = 5$ , making it $(x-5)^2$ .

How do I find the middle term?

Use the perfect square formula! For $(x-a)^2$ , the middle term is always $-2ax$ . Since $a = 5$ , the middle term is $-2(5)x = -10x$ . But the blank asks for the positive coefficient, so it's 10x.

What if the constant term wasn't a perfect square?

If the constant term isn't a perfect square (like 24 or 30), then you cannot write it as a perfect square trinomial. Perfect squares only work when the constant term equals $a^2$ for some whole number $a$ .

Can I check my answer by expanding?

Absolutely! That's the best way to verify. Expand $(x-5)^2$ : $(x-5)(x-5) = x^2 - 5x - 5x + 25 = x^2 - 10x + 25$ . This matches the given form perfectly!

Why does the problem show the middle term as positive 10x in the answer?

The equation format is $x^2 - ? + 25$ , where the blank represents the coefficient of the middle term. Since the actual middle term is $-10x$ , the coefficient is 10x (the minus sign is already shown).

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