Solve Algebra Like Magic. Test Your IQ. Can You Crack It?
Welcome to your ultimate guide on how to solve rational equations step by step. If you've ever felt stuck looking at fractions with variables in the denominator, you're in the right place. We'll start with simple algebra puzzles and build all the way up to quadratic rational equations the kind you meet in Algebra 2, GCSE, and A-level exams. Our main dish today is a clever rational equation with fractions: x/(x-2) + 2/(x+3) = 7/((x-2)(x+3)). We'll solve it three different ways, so you fully understand clearing denominators in equations and how to spot extraneous solutions in rational equations.
Every problem below is broken into multiple methods. This isn't just about getting the answer it's about building a toolkit so you can tackle any rational equations worksheet with confidence.
1. Solve y³ + y³ + y³ = 81
Find y = ?
Method 1 – Combine Like Terms (Easiest)
Method 2 – Factoring Approach
2. Solve x/2 + x/5 = 9
Method 1 – Multiply by LCM (Clearing Denominators)
Method 2 – Common Denominator First
3. Solve 3a² = 75 and ab = 90
ab = 90
Find b² = ?
Method 1 – Find a, then b
Method 2 – Square Product Shortcut (No a needed)
4. Solve 2x² + 5x - 3 = 0
Method 1 – Factoring by Grouping
Method 2 – Quadratic Formula
5. Given a² = 25, ab = 50
ab = 50
b² = ?
Method 1 – Stepwise
a = ±5. If a=5, b=10. If a=-5, b=-10. b²=100.
Method 2 – Square Shortcut
(ab)²=2500 → a²b²=2500 → 25·b²=2500 → b²=100.
6. Solve 4x² = 64
Method 1 – Direct Solve
Divide by 4: x²=16 → x=±4.
Method 2 – Square Root Both Sides
√(4x²)=√64 → 2|x|=8 → |x|=4 → x=±4.
7. Solve x/3 + x/6 = 4
Method 1 – Clear Denominators (LCM 6)
Multiply by 6: 2x + x = 24 → 3x=24 → x=8.
Method 2 – Combine Fractions
2x/6 + x/6 = 4 → 3x/6 = 4 → x/2 = 4 → x=8.
8. Solve x² - 7x + 12 = 0
Method 1 – Factoring
Numbers multiplying to 12, adding to -7: -3 and -4. (x-3)(x-4)=0 → x=3,4.
Method 2 – Quadratic Formula
x = [7 ± √(49-48)]/2 = (7±1)/2 → 4 or 3.
9. Given 2a² = 50, ab = 30
ab = 30
b² = ?
Method 1 – Find a and b
a²=25, a=±5. b=±6. b²=36.
Method 2 – Square Product
(ab)²=900 → 25·b²=900 → b²=36.
10. Solve 5x² + 10x - 15 = 0
Method 1 – Simplify First
Divide by 5: x²+2x-3=0. Factor: (x+3)(x-1)=0 → x=-3,1.
Method 2 – Quadratic Formula Direct
x = [-10 ± √(100+300)]/10 = [-10 ± 20]/10 → 1 or -3.
11. Solve Rational Equations Step by Step: x/(x-2) + 2/(x+3) = 7/((x-2)(x+3))
This is our main rational equation example. It's a perfect practice problem for algebra 2 rational equations because it combines clearing denominators in equations with solving a quadratic rational equation. We'll explore three full methods.
Denominators cannot be zero. Set each to not zero:
x-2 ≠ 0 → x ≠ 2
x+3 ≠ 0 → x ≠ -3
(x-2)(x+3) ≠ 0 confirms both. Always check your final answers against these restrictions.
Method 1 – Multiply by Common Denominator (LCM) — The Pro Approach
This is the most efficient way for solving rational equations. We identify the LCM of all denominators and multiply every term, instantly clearing fractions.
Second term: (x+3) cancels, leaving 2(x-2).
Right side: Everything cancels, leaving just 7.
x² + 5x - 4 = 7
x² + 5x - 11 = 0
Method 2 – Combine Left Side into a Single Fraction
This method builds understanding of fraction addition before clearing denominators. It's excellent for visual learners.
2/(x+3) = 2(x-2) / [(x-2)(x+3)]
Method 3 – Verification & Numeric Check (Build Intuition)
Let's approximate to see if our solutions make sense. √69 is about 8.3066.
Root 1: x = (-5 + 8.3066)/2 ≈ 1.6533.
Plug into original left side: 1.6533/(-0.3467) + 2/4.6533 ≈ -4.768 + 0.430 ≈ -4.338.
Right side: 7/((-0.3467)*4.6533) ≈ 7/(-1.613) ≈ -4.338. Matches!
Root 2: x = (-5 - 8.3066)/2 ≈ -6.6533. Similar check confirms.
This verification step is how you avoid extraneous solutions rational equations traps.
Solve Equations: Zero to Hero
If you've read this far, you've seen rational equations examples ranging from simple to advanced. Let's formalize the process so you can approach any problem confidently. Whether you're working through a rational equations worksheet with answers or preparing for an exam, these steps are your blueprint.
4-Step Framework for Solving Rational Equations
For step by step algebra fractions GCSE / A-level exam prep, practice writing restrictions first it's a common mark to earn. Many students lose points not because they can't solve the quadratic, but because they forget to check for extraneous solutions.
Frequently Asked Questions
Identify the least common denominator (LCD) of all fractions. Multiply every term by that LCD to clear denominators. Solve the resulting equation (usually linear or quadratic). Finally, check that no solution makes any original denominator zero those are extraneous and must be discarded.
An extraneous solution is a mathematically correct answer to the simplified equation that fails in the original equation. This happens when a solution makes a denominator zero. Always substitute your answers back into the original denominators to verify.
Multiplying by the common denominator is the fastest method. It turns a messy fraction equation into a simple polynomial. If you have only one fraction on each side, you can cross-multiply. For multiple terms, clear denominators with the LCM.
Yes, but only when you have a single fraction on each side of the equals sign. For equations with multiple terms, combine into one fraction first, or multiply every term by the common denominator that's the safer, more general method for solving equations with fractions.
You just conquered 11 algebra puzzles from simple cubes to quadratic rational equations. Keep these patterns in your toolkit, and you'll breeze through any algebra 2 rational equations worksheet. Remember: clear denominators, check for extraneous solutions, and practice daily.
JPhilips 
