Solve a(a+4) vs a² in Exam Statistics: Finding Total Class Size

Q: Why do we subtract the number who failed from those who passed?

The problem states "the difference between these numbers is 12" and more students passed than failed. This means passed - failed = 12, so we write $ a(a+4) - a^2 = 12 $.

Q: What if I got a = -3 instead of a = 3?

While $ 4a = 12 $ only gives $ a = 3 $, if you made an error and got $ a = -3 $, remember that number of students must be positive! Negative students don't make sense in real problems.

Q: How do I know which expression represents passed vs failed students?

Read carefully! The problem states "$ a(a+4) $ of students passed" and "$ a^2 $ failed". Always match the algebraic expressions to what they represent in the word problem.

Q: Why don't we need to use the quadratic formula?

After expanding $ a(a+4) - a^2 $, the $ a^2 $ terms cancel out! We get $ 4a = 12 $, which is linear, not quadratic. Sometimes algebra problems simplify more than expected!

Q: Should I always check my answer by substituting back?

Yes, always! With $ a = 3 $: 21 students passed, 9 failed, difference is $ 21 - 9 = 12 $ ✓ and total is $ 21 + 9 = 30 $ students.

Algebraic Expressions with Difference Equations

A maths class takes an exam.

$a(a+4)$ of students passed, while $a^2$ failed the exam.

More students passed than failed and the difference between these numbers is 12.

How many students are in the class?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Step-by-step solution

To solve the problem, we start by establishing the given conditions:

More students passed than failed, and the difference in their numbers is 12.
This translates into the equation: $a(a+4) - a^2 = 12$ .
Expanding and simplifying the left side, we have: $a^2 + 4a - a^2 = 12$ .
Simplify further: $4a = 12$ .
By dividing both sides by 4, we find: $a = 3$ .

Using $a = 3$ to determine the number of students:

Number of students who passed: $3(3 + 4) = 21$ .
Number of students who failed: $3^2 = 9$ .
Total number of students: $21 + 9 = 30$ .

Therefore, the total number of students in the class is $\boxed{30}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Setup: Write difference equation from word problem constraints
Technique: Expand $a(a+4) - a^2 = 4a = 12$
Check: Verify 21 passed, 9 failed gives difference of 12 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand algebraic expressions properly
Don't leave $a(a+4) - a^2$ unexpanded = can't solve for a! Students often skip the expansion step and get stuck. Always expand $a(a+4)$ to $a^2 + 4a$ first, then subtract $a^2$ to get $4a$ .

Practice Quiz

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$$ 5x=1 $$

What is the value of x?

FAQ

Everything you need to know about this question

Why do we subtract the number who failed from those who passed?

The problem states "the difference between these numbers is 12" and more students passed than failed. This means passed - failed = 12, so we write $a(a+4) - a^2 = 12$ .

What if I got a = -3 instead of a = 3?

While $4a = 12$ only gives $a = 3$ , if you made an error and got $a = -3$ , remember that number of students must be positive! Negative students don't make sense in real problems.

How do I know which expression represents passed vs failed students?

Read carefully! The problem states " $a(a+4)$ of students passed" and " $a^2$ failed". Always match the algebraic expressions to what they represent in the word problem.

Why don't we need to use the quadratic formula?

After expanding $a(a+4) - a^2$ , the $a^2$ terms cancel out! We get $4a = 12$ , which is linear, not quadratic. Sometimes algebra problems simplify more than expected!

Should I always check my answer by substituting back?

Yes, always! With $a = 3$ : 21 students passed, 9 failed, difference is $21 - 9 = 12$ ✓ and total is $21 + 9 = 30$ students.

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