Complete the Square: Transform x²+9x+24 into (x+5)²

Q: Why can't I just look at the constant terms 24 and 25?

You need to consider both the linear and constant terms! The difference isn't just 25 - 24 = 1. When you expand $ (x+5)^2 $, you get $ x^2 + 10x + 25 $, so you need x + 1 to transform the 9x and 24.

Q: How do I know which perfect square form to aim for?

The problem tells you! It asks what to add so that $ y = (x+5)^2 $. Your job is to expand this target form and compare it with the given expression.

Q: What if I get a different answer when I expand?

Double-check your expansion using $ (a+b)^2 = a^2 + 2ab + b^2 $. For $ (x+5)^2 $: a = x, b = 5, so you get $ x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25 $.

Q: Can I solve this by completing the square instead?

That would be much harder! This problem gives you the target form $ (x+5)^2 $. Simply expand it and compare - no need to complete the square from scratch.

Q: How do I check if x + 1 is really the right answer?

Add it to the original: $ (x^2 + 9x + 24) + (x + 1) = x^2 + 10x + 25 $. This should equal $ (x+5)^2 = x^2 + 10x + 25 $. Perfect match! ✓

Algebraic Manipulation with Perfect Square Forms

$y=x^2+9x+24$

Which expression should be added to y so that:

$y=(x+5)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 What expression should be added for this equation to be correct?

00:10 Let's apply the short multiplication formulas to expand the parentheses.

00:16 Next, we will calculate the squares and multiplications carefully.

00:24 Be sure to notice any differences as we go through the steps.

00:36 And 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

$y=x^2+9x+24$

Which expression should be added to y so that:

$y=(x+5)^2$

Step-by-step solution

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

Step 1: Expand $(x+5)^2$ using the formula for the square of a sum.
Step 2: Compare the expanded expression with $x^2 + 9x + 24$ .
Step 3: Determine what additional expression is needed to make the two expressions equal.

Let's go through these steps in detail:
Step 1: First, we expand $(x+5)^2$ using the formula $(a+b)^2 = a^2 + 2ab + b^2$ :
$(x+5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2 = x^2 + 10x + 25$ .

Step 2: Now, compare the expanded expression $x^2 + 10x + 25$ with the given $x^2 + 9x + 24$ .
We notice that the linear term 10x needs to be replaced or adjusted with 9x, and the constant 25 with 24.

Step 3: To make the expressions equal, find the difference in linear and constant terms:
$10x + 25$ must equal $9x + 24 + k$ (where $k$ is what we need to add):
Equating them, we get $10x + 25 = 9x + 24 + k$ .
Solve for $k$ :
$10x + 25 - 9x - 24 = k$ .
$x + 1 = k$ .

Therefore, the expression that should be added is $x+1$ .

Final Answer

$x+1$

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: $(x+a)^2 = x^2 + 2ax + a^2$ for perfect squares
Comparison Method: Expand $(x+5)^2 = x^2 + 10x + 25$ and compare coefficients
Verification: Add $x+1$ to original: $x^2+9x+24+(x+1) = x^2+10x+25$ ✓

Common Mistakes

Avoid these frequent errors

Comparing expressions without proper expansion
Don't try to guess what to add without first expanding $(x+5)^2$ completely! This leads to random guessing and wrong answers. Always expand the target form first, then systematically compare each term to find the exact difference.

Practice Quiz

Test your knowledge with interactive questions

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

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

Why can't I just look at the constant terms 24 and 25?

You need to consider both the linear and constant terms! The difference isn't just 25 - 24 = 1. When you expand $(x+5)^2$ , you get $x^2 + 10x + 25$ , so you need x + 1 to transform the 9x and 24.

How do I know which perfect square form to aim for?

The problem tells you! It asks what to add so that $y = (x+5)^2$ . Your job is to expand this target form and compare it with the given expression.

What if I get a different answer when I expand?

Double-check your expansion using $(a+b)^2 = a^2 + 2ab + b^2$ . For $(x+5)^2$ : a = x, b = 5, so you get $x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25$ .

Can I solve this by completing the square instead?

That would be much harder! This problem gives you the target form $(x+5)^2$ . Simply expand it and compare - no need to complete the square from scratch.

How do I check if x + 1 is really the right answer?

Add it to the original: $(x^2 + 9x + 24) + (x + 1) = x^2 + 10x + 25$ . This should equal $(x+5)^2 = x^2 + 10x + 25$ . Perfect match! ✓

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