Find the Missing Terms: Completing the Quadratic Expression

Q: How do I know which formula to use for perfect squares?

Look for the pattern $ (something)^2 $ on one side. This tells you to use the perfect square formula: $ (a + b)^2 = a^2 + 2ab + b^2 $.

Q: Why is the middle term 16x and not just 16?

The middle term comes from $ 2ab $ where $ a = 2x $ and $ b = 4 $. So $ 2(2x)(4) = 16x $, not just 16!

Q: What if I get the wrong constant term?

Remember that the constant term is $ b^2 $. If you found $ b = 4 $, then $ b^2 = 16 $, not 4!

Q: How can I check my answer quickly?

Expand your completed expression: $ (2x + 4)^2 $. If it matches $ 4x^2 + 16x + 16 $, you're correct!

Q: What does the 2 in front of 2x tell me?

The coefficient 2 in $ 2x $ is your first term $ a $. When you square it, you get $ (2x)^2 = 4x^2 $, which matches the first term in the expansion.

Perfect Square Trinomials with Missing Terms

Complete what is missing:

$(2x+?\rparen^2=4x^2+16x+?$

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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 Set B as unknown

00:12 Use the shortened multiplication formulas to open the parentheses

00:37 Compare coefficients to find B

00:44 Isolate B

00:53 This is the unknown B

01:02 Calculate B(4) squared and substitute

01:05 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

Complete what is missing:

$(2x+?\rparen^2=4x^2+16x+?$

Step-by-step solution

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

Step 1: Identify the terms in the binomial $(2x + ?)^2$ .
Step 2: Apply the formula $(a + b)^2 = a^2 + 2ab + b^2$ to ascertain missing values.
Step 3: Compare with given expanded form and determine the missing terms.

Now, let's work through each step:
Step 1: We have $(2x + ?)^2$ . Here, $a = 2x$ and the missing term replaces ?. Assume it is $b$ .
Step 2: Use the formula $(a + b)^2 = a^2 + 2ab + b^2$ . Replace $a$ with $2x$ giving $a^2 = (2x)^2 = 4x^2$ .
Step 3: Comparison with expanded result: We need $2ab$ and $b^2$ such that $(4x^2 + 2ab + b^2 = 4x^2 + 16x + ?$ ). For $2ab = 16x$ , $2(2x)(b) = 16x$ implies $b = 4$ . Substituting $b = 4$ , we calculate: $b^2 = 4^2 = 16$ .

Therefore, the solution to the problem is the missing numbers are 4, 16.

Final Answer

4, 16

Key Points to Remember

Essential concepts to master this topic

Formula: Use $(a + b)^2 = a^2 + 2ab + b^2$ to expand
Technique: From $2ab = 16x$ , solve $2(2x)(b) = 16x$ to get b = 4
Check: Verify $(2x + 4)^2 = 4x^2 + 16x + 16$ matches pattern ✓

Common Mistakes

Avoid these frequent errors

Incorrectly identifying the middle term coefficient
Don't assume the missing term equals the coefficient of the middle term (16) = wrong answer 16, 256! The middle term 16x comes from 2ab, not just b. Always solve 2ab = 16x step by step to find the actual missing term.

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

How do I know which formula to use for perfect squares?

Look for the pattern $(something)^2$ on one side. This tells you to use the perfect square formula: $(a + b)^2 = a^2 + 2ab + b^2$ .

Why is the middle term 16x and not just 16?

The middle term comes from $2ab$ where $a = 2x$ and $b = 4$ . So $2(2x)(4) = 16x$ , not just 16!

What if I get the wrong constant term?

Remember that the constant term is $b^2$ . If you found $b = 4$ , then $b^2 = 16$ , not 4!

How can I check my answer quickly?

Expand your completed expression: $(2x + 4)^2$ . If it matches $4x^2 + 16x + 16$ , you're correct!

What does the 2 in front of 2x tell me?

The coefficient 2 in $2x$ is your first term $a$ . When you square it, you get $(2x)^2 = 4x^2$ , which matches the first term in the expansion.

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