Determine Variables A, B, and C for the Equation: (A/X) + (BX/2) = ((2X+3)²)/X - C

Q: Why do we need to expand the quadratic first?

Expanding $ (2X+3)^2 $ reveals all the individual terms we need to match. Without expansion, you can't see that the coefficient of X is 12 or that the constant term is 9.

Q: How do I know which terms to compare?

Match terms with the same powers of X: $ \frac{1}{X} $ terms together, X terms together, and constants together. This gives you three separate equations to solve.

Q: What if I get different values when I check my answer?

Go back and check your expansion and simplification steps. The most common errors happen when expanding $ (2X+3)^2 $ or when dividing $ \frac{4X^2 + 12X + 9}{X} $.

Q: Why does the equation work for all values of X?

This is an identity - it's true for every value of X because we're finding the exact coefficients that make both sides identical. That's why coefficient comparison works!

Algebraic Equations with Coefficient Comparison

$\frac{A}{X}+\frac{BX}{2}=\frac{(2X+3)^2}{X}-C$

Calculate the values of A, B, and C so that the equation is satisfied.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find A,B,C

00:03 Multiply by the common denominator to eliminate fractions

00:44 Reduce what's possible

00:56 Use shortened multiplication formulas to open the parentheses

01:19 Move C to the left side

01:24 Calculate the squares and products

01:40 Open parentheses properly, multiply by each factor

01:50 Equate each coefficient on the right side to the coefficient on the left side

01:56 This is the solution for B

02:07 Factor 24 into factors 2 and 12

02:16 This is the solution for C

02:23 Factor 18 into factors 9 and 2

02:27 This is the solution for A

02:30 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

$\frac{A}{X}+\frac{BX}{2}=\frac{(2X+3)^2}{X}-C$

Calculate the values of A, B, and C so that the equation is satisfied.

Step-by-step solution

To solve this problem, we will simplify both sides of the given equation:

Given equation:

$\frac{A}{X} + \frac{BX}{2} = \frac{(2X+3)^2}{X} - C$ .

First, expand the quadratic expression:

$(2X+3)^2 = (2X+3)(2X+3) = 4X^2 + 6X + 6X + 9 = 4X^2 + 12X + 9$ .

Substitute this back into the equation:

$\frac{A}{X} + \frac{BX}{2} = \frac{4X^2 + 12X + 9}{X} - C$ .

Simplify the right-hand side:

$\frac{4X^2 + 12X + 9}{X} = 4X + \frac{12X}{X} + \frac{9}{X} = 4X + 12 + \frac{9}{X}$ .

The equation now becomes:

$\frac{A}{X} + \frac{BX}{2} = 4X + 12 + \frac{9}{X} - C$ .

For the equation to hold true for all values of $X$ , equate corresponding terms:

$\frac{A}{X} = \frac{9}{X} \Rightarrow A = 9$ .
$\frac{BX}{2} = 4X \Rightarrow B = 8$ .
For constant terms: $12 - C = 0 \Rightarrow C = 12$ .

Therefore, the values are $A = 9$ , $B = 8$ , and $C = 12$ .

The correct answer is: $A = 9, B = 8, C = 12$ .

Final Answer

$A=9,B=8,C=12$

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: Expand quadratic expressions completely before simplifying fractions
Technique: Compare coefficients: $\frac{BX}{2} = 4X$ gives $B = 8$
Check: Substitute A=9, B=8, C=12 back into original equation ✓

Common Mistakes

Avoid these frequent errors

Not expanding the quadratic expression completely
Don't skip expanding $(2X+3)^2$ = wrong coefficients! Students often rush and miss terms like the middle term 12X. Always expand step-by-step: $(2X+3)^2 = 4X^2 + 12X + 9$ .

Practice Quiz

Test your knowledge with interactive questions

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

$$ (x+y)^2 $$

FAQ

Everything you need to know about this question

Why do we need to expand the quadratic first?

Expanding $(2X+3)^2$ reveals all the individual terms we need to match. Without expansion, you can't see that the coefficient of X is 12 or that the constant term is 9.

How do I know which terms to compare?

Match terms with the same powers of X: $\frac{1}{X}$ terms together, X terms together, and constants together. This gives you three separate equations to solve.

What if I get different values when I check my answer?

Go back and check your expansion and simplification steps. The most common errors happen when expanding $(2X+3)^2$ or when dividing $\frac{4X^2 + 12X + 9}{X}$ .

Why does the equation work for all values of X?

This is an identity - it's true for every value of X because we're finding the exact coefficients that make both sides identical. That's why coefficient comparison works!

Can I solve this without expanding the right side?

No, you must expand first. The whole point is to get both sides in the same form ( $\frac{A}{X} + BX + C$ ) so you can compare corresponding terms.

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