Practice worksheet for simplifying rational expressions, featuring 30 algebraic problems.
A worksheet titled "Simplifying Rational Expressions—Practice" with 30 numbered problems involving algebraic fractions to be simplified.
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Show Answer Key & Explanations
Step-by-step solution for: Free Printable Simplifying Rational Expressions Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Free Printable Simplifying Rational Expressions Worksheets
Let’s simplify each rational expression one by one. We’ll factor where needed, cancel common factors, and write the simplest form.
---
1. $\frac{8x^2}{12x^4}$
→ Simplify coefficients: $8/12 = 2/3$
→ Subtract exponents for x: $x^{2-4} = x^{-2} = \frac{1}{x^2}$
→ Final: $\boxed{\frac{2}{3x^2}}$
---
2. $\frac{24x^2y}{32xy^3}$
→ Coefficients: $24/32 = 3/4$
→ x: $x^{2-1} = x$
→ y: $y^{1-3} = y^{-2} = \frac{1}{y^2}$
→ Final: $\boxed{\frac{3x}{4y^2}}$
---
3. $\frac{(x+1)^3}{(x+1)^5}$
→ Subtract exponents: $(x+1)^{3-5} = (x+1)^{-2} = \frac{1}{(x+1)^2}$
→ Final: $\boxed{\frac{1}{(x+1)^2}}$
---
4. $\frac{(3x - 2)^6}{(3x - 2)^4}$
→ Subtract exponents: $(3x - 2)^{6-4} = (3x - 2)^2$
→ Final: $\boxed{(3x - 2)^2}$
---
5. $\frac{2n - 3}{3 - 2n}$
→ Notice: $3 - 2n = -(2n - 3)$
→ So: $\frac{2n - 3}{-(2n - 3)} = -1$
→ Final: $\boxed{-1}$
---
6. $\frac{15y^2(y + 3)}{20y^3(y + 3)}$
→ Cancel $(y + 3)$ top and bottom
→ Coefficients: $15/20 = 3/4$
→ y: $y^{2-3} = y^{-1} = \frac{1}{y}$
→ Final: $\boxed{\frac{3}{4y}}$
---
7. $\frac{15y(y + 1)}{45y^2(y + 1)}$
→ Cancel $(y + 1)$
→ Coefficients: $15/45 = 1/3$
→ y: $y^{1-2} = y^{-1} = \frac{1}{y}$
→ Final: $\boxed{\frac{1}{3y}}$
---
8. $\frac{24x^3(4 - x)}{18x(4 - x)}$
→ Cancel $(4 - x)$
→ Coefficients: $24/18 = 4/3$
→ x: $x^{3-1} = x^2$
→ Final: $\boxed{\frac{4x^2}{3}}$
---
9. $\frac{6x^3(2 - x)}{12x(2 - x)}$
→ Cancel $(2 - x)$
→ Coefficients: $6/12 = 1/2$
→ x: $x^{3-1} = x^2$
→ Final: $\boxed{\frac{x^2}{2}}$
---
10. $\frac{12x^3(5 - 2x)}{18x(2x - 5)}$
→ Note: $5 - 2x = -(2x - 5)$
→ So numerator becomes: $12x^3 \cdot (-(2x - 5)) = -12x^3(2x - 5)$
→ Now: $\frac{-12x^3(2x - 5)}{18x(2x - 5)}$ → cancel $(2x - 5)$
→ Coefficients: $-12/18 = -2/3$
→ x: $x^{3-1} = x^2$
→ Final: $\boxed{-\frac{2x^2}{3}}$
---
11. $\frac{a^2 + 6a}{ac + 6c}$
→ Factor numerator: $a(a + 6)$
→ Factor denominator: $c(a + 6)$
→ Cancel $(a + 6)$
→ Left with: $\frac{a}{c}$
→ Final: $\boxed{\frac{a}{c}}$
---
12. $\frac{x^2 + 4x}{3x + 12}$
→ Numerator: $x(x + 4)$
→ Denominator: $3(x + 4)$
→ Cancel $(x + 4)$
→ Left with: $\frac{x}{3}$
→ Final: $\boxed{\frac{x}{3}}$
---
13. $\frac{6 - 8x}{4x^2 - 3x}$
→ Numerator: factor out -2? Or just leave as is for now. Let’s factor both.
Numerator: $2(3 - 4x)$ or $-2(4x - 3)$
Denominator: $x(4x - 3)$
→ So: $\frac{-2(4x - 3)}{x(4x - 3)}$ → cancel $(4x - 3)$
→ Left with: $\frac{-2}{x}$
→ Final: $\boxed{-\frac{2}{x}}$
---
14. $\frac{x^2 + 2x - 15}{x^2 - 10x + 21}$
→ Factor numerator: $(x + 5)(x - 3)$
→ Factor denominator: $(x - 3)(x - 7)$
→ Cancel $(x - 3)$
→ Left with: $\frac{x + 5}{x - 7}$
→ Final: $\boxed{\frac{x + 5}{x - 7}}$
---
15. $\frac{x^2 - 2x - 3}{x^2 - 5x - 6}$
→ Numerator: $(x - 3)(x + 1)$
→ Denominator: $(x - 6)(x + 1)$
→ Cancel $(x + 1)$
→ Left with: $\frac{x - 3}{x - 6}$
→ Final: $\boxed{\frac{x - 3}{x - 6}}$
---
16. $\frac{x^2 - 7x + 12}{x^2 - x - 12}$
→ Numerator: $(x - 3)(x - 4)$
→ Denominator: $(x - 4)(x + 3)$
→ Cancel $(x - 4)$
→ Left with: $\frac{x - 3}{x + 3}$
→ Final: $\boxed{\frac{x - 3}{x + 3}}$
---
17. $\frac{x^2 - 3x - 10}{x^2 - 6x + 5}$
→ Numerator: $(x - 5)(x + 2)$
→ Denominator: $(x - 5)(x - 1)$
→ Cancel $(x - 5)$
→ Left with: $\frac{x + 2}{x - 1}$
→ Final: $\boxed{\frac{x + 2}{x - 1}}$
---
18. $\frac{x^2 - 8x + 16}{x^2 - 9x + 20}$
→ Numerator: $(x - 4)^2$
→ Denominator: $(x - 4)(x - 5)$
→ Cancel one $(x - 4)$
→ Left with: $\frac{x - 4}{x - 5}$
→ Final: $\boxed{\frac{x - 4}{x - 5}}$
---
19. $\frac{x^2 - 7x + 10}{x^2 - 4x - 5}$
→ Numerator: $(x - 2)(x - 5)$
→ Denominator: $(x - 5)(x + 1)$
→ Cancel $(x - 5)$
→ Left with: $\frac{x - 2}{x + 1}$
→ Final: $\boxed{\frac{x - 2}{x + 1}}$
---
20. $\frac{x^2 + 6x + 9}{x^2 + 2x - 3}$
→ Numerator: $(x + 3)^2$
→ Denominator: $(x + 3)(x - 1)$
→ Cancel one $(x + 3)$
→ Left with: $\frac{x + 3}{x - 1}$
→ Final: $\boxed{\frac{x + 3}{x - 1}}$
---
21. $\frac{x^2 - 2x - 24}{16 - x^2}$
→ Numerator: $(x - 6)(x + 4)$
→ Denominator: $16 - x^2 = -(x^2 - 16) = -(x - 4)(x + 4)$
→ So: $\frac{(x - 6)(x + 4)}{-(x - 4)(x + 4)}$ → cancel $(x + 4)$
→ Left with: $\frac{x - 6}{-(x - 4)} = -\frac{x - 6}{x - 4}$
→ Can also write as $\frac{6 - x}{x - 4}$ — either is fine, but let’s keep it as $-\frac{x - 6}{x - 4}$
→ Final: $\boxed{-\frac{x - 6}{x - 4}}$
---
22. $\frac{9 - y^2}{y^2 + 2y - 15}$
→ Numerator: $9 - y^2 = -(y^2 - 9) = -(y - 3)(y + 3)$
→ Denominator: $(y + 5)(y - 3)$
→ So: $\frac{-(y - 3)(y + 3)}{(y + 5)(y - 3)}$ → cancel $(y - 3)$
→ Left with: $\frac{-(y + 3)}{y + 5} = -\frac{y + 3}{y + 5}$
→ Final: $\boxed{-\frac{y + 3}{y + 5}}$
---
23. $\frac{3x^2 + 18x^2 - 21x}{x^3 + 5x^2 - 14x}$
Wait — typo? First term in numerator: $3x^2 + 18x^2 = 21x^2$, so numerator is $21x^2 - 21x$
Factor: $21x(x - 1)$
Denominator: $x(x^2 + 5x - 14) = x(x + 7)(x - 2)$
So: $\frac{21x(x - 1)}{x(x + 7)(x - 2)}$ → cancel x
→ Left with: $\frac{21(x - 1)}{(x + 7)(x - 2)}$
→ Cannot simplify further
→ Final: $\boxed{\frac{21(x - 1)}{(x + 7)(x - 2)}}$
*(Note: If original was meant to be $3x^3 + 18x^2 - 21x$, then we’d factor differently — but based on what’s written, this is correct.)*
---
24. $\frac{2x^2 - 18x}{4x^3 - 32x^2 - 36x}$
→ Numerator: $2x(x - 9)$
→ Denominator: factor out 4x? Wait: all terms divisible by 2x? Let’s see:
Denominator: $4x^3 - 32x^2 - 36x = 4x(x^2 - 8x - 9)$
Now factor quadratic: $x^2 - 8x - 9 = (x - 9)(x + 1)$
So denominator: $4x(x - 9)(x + 1)$
Numerator: $2x(x - 9)$
→ So: $\frac{2x(x - 9)}{4x(x - 9)(x + 1)}$ → cancel $2x(x - 9)$
Left with: $\frac{1}{2(x + 1)}$
→ Final: $\boxed{\frac{1}{2(x + 1)}}$
---
25. $\frac{5x^2 - 11x + 2}{5x^2 - 7x - 6}$
→ Factor numerator: Look for two numbers that multiply to $5*2=10$, add to -11 → -10 and -1
Split middle: $5x^2 - 10x - x + 2 = 5x(x - 2) -1(x - 2) = (5x - 1)(x - 2)$
→ Denominator: $5x^2 - 7x - 6$ → multiply 5*(-6)= -30, find pair adding to -7 → -10 and 3
Split: $5x^2 - 10x + 3x - 6 = 5x(x - 2) + 3(x - 2) = (5x + 3)(x - 2)$
→ So: $\frac{(5x - 1)(x - 2)}{(5x + 3)(x - 2)}$ → cancel $(x - 2)$
→ Left with: $\frac{5x - 1}{5x + 3}$
→ Final: $\boxed{\frac{5x - 1}{5x + 3}}$
---
26. $\frac{2x^2 - 7x + 3}{2x^2 + 9x - 5}$
→ Numerator: $2x^2 - 7x + 3$ → factors: (2x - 1)(x - 3)
Check: 2x*x=2x², 2x*(-3)= -6x, -1*x= -x, total -7x; (-1)*(-3)=3 ✔️
→ Denominator: $2x^2 + 9x - 5$ → factors: (2x - 1)(x + 5)
Check: 2x*x=2x², 2x*5=10x, -1*x=-x, total 9x; (-1)*5=-5 ✔️
→ So: $\frac{(2x - 1)(x - 3)}{(2x - 1)(x + 5)}$ → cancel $(2x - 1)$
→ Left with: $\frac{x - 3}{x + 5}$
→ Final: $\boxed{\frac{x - 3}{x + 5}}$
---
27. $\frac{x^2 + 2x - 24}{12 - 4x - x^2}$
→ Numerator: $(x + 6)(x - 4)$
→ Denominator: rewrite as $-x^2 - 4x + 12 = -(x^2 + 4x - 12) = -(x + 6)(x - 2)$
→ So: $\frac{(x + 6)(x - 4)}{-(x + 6)(x - 2)}$ → cancel $(x + 6)$
→ Left with: $\frac{x - 4}{-(x - 2)} = -\frac{x - 4}{x - 2}$
→ Final: $\boxed{-\frac{x - 4}{x - 2}}$
---
28. $\frac{2x^2 - 8}{x^2 + 4x - 12}$
→ Numerator: $2(x^2 - 4) = 2(x - 2)(x + 2)$
→ Denominator: $(x + 6)(x - 2)$
→ So: $\frac{2(x - 2)(x + 2)}{(x + 6)(x - 2)}$ → cancel $(x - 2)$
→ Left with: $\frac{2(x + 2)}{x + 6}$
→ Final: $\boxed{\frac{2(x + 2)}{x + 6}}$
---
29. $\frac{2x^2 - 9x + 7}{x^2 + 2x - 3}$
→ Numerator: $2x^2 - 9x + 7$ → factors: (2x - 7)(x - 1)? Check: 2x*x=2x², 2x*(-1)= -2x, -7*x= -7x, total -9x; (-7)*(-1)=7 ✔️
→ Denominator: $(x + 3)(x - 1)$
→ So: $\frac{(2x - 7)(x - 1)}{(x + 3)(x - 1)}$ → cancel $(x - 1)$
→ Left with: $\frac{2x - 7}{x + 3}$
→ Final: $\boxed{\frac{2x - 7}{x + 3}}$
---
30. $\frac{8x^2 - 14x + 6}{4x - 3}$
→ Try factoring numerator: $8x^2 - 14x + 6$
Divide by 2: $2(4x^2 - 7x + 3)$
Factor inside: $4x^2 - 7x + 3 = (4x - 3)(x - 1)$? Check: 4x*x=4x², 4x*(-1)= -4x, -3*x= -3x, total -7x; (-3)*(-1)=3 ✔️
So numerator: $2(4x - 3)(x - 1)$
Denominator: $4x - 3$
→ So: $\frac{2(4x - 3)(x - 1)}{4x - 3}$ → cancel $(4x - 3)$
→ Left with: $2(x - 1)$
→ Final: $\boxed{2(x - 1)}$
---
Final Answer:
1. $\frac{2}{3x^2}$
2. $\frac{3x}{4y^2}$
3. $\frac{1}{(x+1)^2}$
4. $(3x - 2)^2$
5. $-1$
6. $\frac{3}{4y}$
7. $\frac{1}{3y}$
8. $\frac{4x^2}{3}$
9. $\frac{x^2}{2}$
10. $-\frac{2x^2}{3}$
11. $\frac{a}{c}$
12. $\frac{x}{3}$
13. $-\frac{2}{x}$
14. $\frac{x + 5}{x - 7}$
15. $\frac{x - 3}{x - 6}$
16. $\frac{x - 3}{x + 3}$
17. $\frac{x + 2}{x - 1}$
18. $\frac{x - 4}{x - 5}$
19. $\frac{x - 2}{x + 1}$
20. $\frac{x + 3}{x - 1}$
21. $-\frac{x - 6}{x - 4}$
22. $-\frac{y + 3}{y + 5}$
23. $\frac{21(x - 1)}{(x + 7)(x - 2)}$
24. $\frac{1}{2(x + 1)}$
25. $\frac{5x - 1}{5x + 3}$
26. $\frac{x - 3}{x + 5}$
27. $-\frac{x - 4}{x - 2}$
28. $\frac{2(x + 2)}{x + 6}$
29. $\frac{2x - 7}{x + 3}$
30. $2(x - 1)$
---
1. $\frac{8x^2}{12x^4}$
→ Simplify coefficients: $8/12 = 2/3$
→ Subtract exponents for x: $x^{2-4} = x^{-2} = \frac{1}{x^2}$
→ Final: $\boxed{\frac{2}{3x^2}}$
---
2. $\frac{24x^2y}{32xy^3}$
→ Coefficients: $24/32 = 3/4$
→ x: $x^{2-1} = x$
→ y: $y^{1-3} = y^{-2} = \frac{1}{y^2}$
→ Final: $\boxed{\frac{3x}{4y^2}}$
---
3. $\frac{(x+1)^3}{(x+1)^5}$
→ Subtract exponents: $(x+1)^{3-5} = (x+1)^{-2} = \frac{1}{(x+1)^2}$
→ Final: $\boxed{\frac{1}{(x+1)^2}}$
---
4. $\frac{(3x - 2)^6}{(3x - 2)^4}$
→ Subtract exponents: $(3x - 2)^{6-4} = (3x - 2)^2$
→ Final: $\boxed{(3x - 2)^2}$
---
5. $\frac{2n - 3}{3 - 2n}$
→ Notice: $3 - 2n = -(2n - 3)$
→ So: $\frac{2n - 3}{-(2n - 3)} = -1$
→ Final: $\boxed{-1}$
---
6. $\frac{15y^2(y + 3)}{20y^3(y + 3)}$
→ Cancel $(y + 3)$ top and bottom
→ Coefficients: $15/20 = 3/4$
→ y: $y^{2-3} = y^{-1} = \frac{1}{y}$
→ Final: $\boxed{\frac{3}{4y}}$
---
7. $\frac{15y(y + 1)}{45y^2(y + 1)}$
→ Cancel $(y + 1)$
→ Coefficients: $15/45 = 1/3$
→ y: $y^{1-2} = y^{-1} = \frac{1}{y}$
→ Final: $\boxed{\frac{1}{3y}}$
---
8. $\frac{24x^3(4 - x)}{18x(4 - x)}$
→ Cancel $(4 - x)$
→ Coefficients: $24/18 = 4/3$
→ x: $x^{3-1} = x^2$
→ Final: $\boxed{\frac{4x^2}{3}}$
---
9. $\frac{6x^3(2 - x)}{12x(2 - x)}$
→ Cancel $(2 - x)$
→ Coefficients: $6/12 = 1/2$
→ x: $x^{3-1} = x^2$
→ Final: $\boxed{\frac{x^2}{2}}$
---
10. $\frac{12x^3(5 - 2x)}{18x(2x - 5)}$
→ Note: $5 - 2x = -(2x - 5)$
→ So numerator becomes: $12x^3 \cdot (-(2x - 5)) = -12x^3(2x - 5)$
→ Now: $\frac{-12x^3(2x - 5)}{18x(2x - 5)}$ → cancel $(2x - 5)$
→ Coefficients: $-12/18 = -2/3$
→ x: $x^{3-1} = x^2$
→ Final: $\boxed{-\frac{2x^2}{3}}$
---
11. $\frac{a^2 + 6a}{ac + 6c}$
→ Factor numerator: $a(a + 6)$
→ Factor denominator: $c(a + 6)$
→ Cancel $(a + 6)$
→ Left with: $\frac{a}{c}$
→ Final: $\boxed{\frac{a}{c}}$
---
12. $\frac{x^2 + 4x}{3x + 12}$
→ Numerator: $x(x + 4)$
→ Denominator: $3(x + 4)$
→ Cancel $(x + 4)$
→ Left with: $\frac{x}{3}$
→ Final: $\boxed{\frac{x}{3}}$
---
13. $\frac{6 - 8x}{4x^2 - 3x}$
→ Numerator: factor out -2? Or just leave as is for now. Let’s factor both.
Numerator: $2(3 - 4x)$ or $-2(4x - 3)$
Denominator: $x(4x - 3)$
→ So: $\frac{-2(4x - 3)}{x(4x - 3)}$ → cancel $(4x - 3)$
→ Left with: $\frac{-2}{x}$
→ Final: $\boxed{-\frac{2}{x}}$
---
14. $\frac{x^2 + 2x - 15}{x^2 - 10x + 21}$
→ Factor numerator: $(x + 5)(x - 3)$
→ Factor denominator: $(x - 3)(x - 7)$
→ Cancel $(x - 3)$
→ Left with: $\frac{x + 5}{x - 7}$
→ Final: $\boxed{\frac{x + 5}{x - 7}}$
---
15. $\frac{x^2 - 2x - 3}{x^2 - 5x - 6}$
→ Numerator: $(x - 3)(x + 1)$
→ Denominator: $(x - 6)(x + 1)$
→ Cancel $(x + 1)$
→ Left with: $\frac{x - 3}{x - 6}$
→ Final: $\boxed{\frac{x - 3}{x - 6}}$
---
16. $\frac{x^2 - 7x + 12}{x^2 - x - 12}$
→ Numerator: $(x - 3)(x - 4)$
→ Denominator: $(x - 4)(x + 3)$
→ Cancel $(x - 4)$
→ Left with: $\frac{x - 3}{x + 3}$
→ Final: $\boxed{\frac{x - 3}{x + 3}}$
---
17. $\frac{x^2 - 3x - 10}{x^2 - 6x + 5}$
→ Numerator: $(x - 5)(x + 2)$
→ Denominator: $(x - 5)(x - 1)$
→ Cancel $(x - 5)$
→ Left with: $\frac{x + 2}{x - 1}$
→ Final: $\boxed{\frac{x + 2}{x - 1}}$
---
18. $\frac{x^2 - 8x + 16}{x^2 - 9x + 20}$
→ Numerator: $(x - 4)^2$
→ Denominator: $(x - 4)(x - 5)$
→ Cancel one $(x - 4)$
→ Left with: $\frac{x - 4}{x - 5}$
→ Final: $\boxed{\frac{x - 4}{x - 5}}$
---
19. $\frac{x^2 - 7x + 10}{x^2 - 4x - 5}$
→ Numerator: $(x - 2)(x - 5)$
→ Denominator: $(x - 5)(x + 1)$
→ Cancel $(x - 5)$
→ Left with: $\frac{x - 2}{x + 1}$
→ Final: $\boxed{\frac{x - 2}{x + 1}}$
---
20. $\frac{x^2 + 6x + 9}{x^2 + 2x - 3}$
→ Numerator: $(x + 3)^2$
→ Denominator: $(x + 3)(x - 1)$
→ Cancel one $(x + 3)$
→ Left with: $\frac{x + 3}{x - 1}$
→ Final: $\boxed{\frac{x + 3}{x - 1}}$
---
21. $\frac{x^2 - 2x - 24}{16 - x^2}$
→ Numerator: $(x - 6)(x + 4)$
→ Denominator: $16 - x^2 = -(x^2 - 16) = -(x - 4)(x + 4)$
→ So: $\frac{(x - 6)(x + 4)}{-(x - 4)(x + 4)}$ → cancel $(x + 4)$
→ Left with: $\frac{x - 6}{-(x - 4)} = -\frac{x - 6}{x - 4}$
→ Can also write as $\frac{6 - x}{x - 4}$ — either is fine, but let’s keep it as $-\frac{x - 6}{x - 4}$
→ Final: $\boxed{-\frac{x - 6}{x - 4}}$
---
22. $\frac{9 - y^2}{y^2 + 2y - 15}$
→ Numerator: $9 - y^2 = -(y^2 - 9) = -(y - 3)(y + 3)$
→ Denominator: $(y + 5)(y - 3)$
→ So: $\frac{-(y - 3)(y + 3)}{(y + 5)(y - 3)}$ → cancel $(y - 3)$
→ Left with: $\frac{-(y + 3)}{y + 5} = -\frac{y + 3}{y + 5}$
→ Final: $\boxed{-\frac{y + 3}{y + 5}}$
---
23. $\frac{3x^2 + 18x^2 - 21x}{x^3 + 5x^2 - 14x}$
Wait — typo? First term in numerator: $3x^2 + 18x^2 = 21x^2$, so numerator is $21x^2 - 21x$
Factor: $21x(x - 1)$
Denominator: $x(x^2 + 5x - 14) = x(x + 7)(x - 2)$
So: $\frac{21x(x - 1)}{x(x + 7)(x - 2)}$ → cancel x
→ Left with: $\frac{21(x - 1)}{(x + 7)(x - 2)}$
→ Cannot simplify further
→ Final: $\boxed{\frac{21(x - 1)}{(x + 7)(x - 2)}}$
*(Note: If original was meant to be $3x^3 + 18x^2 - 21x$, then we’d factor differently — but based on what’s written, this is correct.)*
---
24. $\frac{2x^2 - 18x}{4x^3 - 32x^2 - 36x}$
→ Numerator: $2x(x - 9)$
→ Denominator: factor out 4x? Wait: all terms divisible by 2x? Let’s see:
Denominator: $4x^3 - 32x^2 - 36x = 4x(x^2 - 8x - 9)$
Now factor quadratic: $x^2 - 8x - 9 = (x - 9)(x + 1)$
So denominator: $4x(x - 9)(x + 1)$
Numerator: $2x(x - 9)$
→ So: $\frac{2x(x - 9)}{4x(x - 9)(x + 1)}$ → cancel $2x(x - 9)$
Left with: $\frac{1}{2(x + 1)}$
→ Final: $\boxed{\frac{1}{2(x + 1)}}$
---
25. $\frac{5x^2 - 11x + 2}{5x^2 - 7x - 6}$
→ Factor numerator: Look for two numbers that multiply to $5*2=10$, add to -11 → -10 and -1
Split middle: $5x^2 - 10x - x + 2 = 5x(x - 2) -1(x - 2) = (5x - 1)(x - 2)$
→ Denominator: $5x^2 - 7x - 6$ → multiply 5*(-6)= -30, find pair adding to -7 → -10 and 3
Split: $5x^2 - 10x + 3x - 6 = 5x(x - 2) + 3(x - 2) = (5x + 3)(x - 2)$
→ So: $\frac{(5x - 1)(x - 2)}{(5x + 3)(x - 2)}$ → cancel $(x - 2)$
→ Left with: $\frac{5x - 1}{5x + 3}$
→ Final: $\boxed{\frac{5x - 1}{5x + 3}}$
---
26. $\frac{2x^2 - 7x + 3}{2x^2 + 9x - 5}$
→ Numerator: $2x^2 - 7x + 3$ → factors: (2x - 1)(x - 3)
Check: 2x*x=2x², 2x*(-3)= -6x, -1*x= -x, total -7x; (-1)*(-3)=3 ✔️
→ Denominator: $2x^2 + 9x - 5$ → factors: (2x - 1)(x + 5)
Check: 2x*x=2x², 2x*5=10x, -1*x=-x, total 9x; (-1)*5=-5 ✔️
→ So: $\frac{(2x - 1)(x - 3)}{(2x - 1)(x + 5)}$ → cancel $(2x - 1)$
→ Left with: $\frac{x - 3}{x + 5}$
→ Final: $\boxed{\frac{x - 3}{x + 5}}$
---
27. $\frac{x^2 + 2x - 24}{12 - 4x - x^2}$
→ Numerator: $(x + 6)(x - 4)$
→ Denominator: rewrite as $-x^2 - 4x + 12 = -(x^2 + 4x - 12) = -(x + 6)(x - 2)$
→ So: $\frac{(x + 6)(x - 4)}{-(x + 6)(x - 2)}$ → cancel $(x + 6)$
→ Left with: $\frac{x - 4}{-(x - 2)} = -\frac{x - 4}{x - 2}$
→ Final: $\boxed{-\frac{x - 4}{x - 2}}$
---
28. $\frac{2x^2 - 8}{x^2 + 4x - 12}$
→ Numerator: $2(x^2 - 4) = 2(x - 2)(x + 2)$
→ Denominator: $(x + 6)(x - 2)$
→ So: $\frac{2(x - 2)(x + 2)}{(x + 6)(x - 2)}$ → cancel $(x - 2)$
→ Left with: $\frac{2(x + 2)}{x + 6}$
→ Final: $\boxed{\frac{2(x + 2)}{x + 6}}$
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29. $\frac{2x^2 - 9x + 7}{x^2 + 2x - 3}$
→ Numerator: $2x^2 - 9x + 7$ → factors: (2x - 7)(x - 1)? Check: 2x*x=2x², 2x*(-1)= -2x, -7*x= -7x, total -9x; (-7)*(-1)=7 ✔️
→ Denominator: $(x + 3)(x - 1)$
→ So: $\frac{(2x - 7)(x - 1)}{(x + 3)(x - 1)}$ → cancel $(x - 1)$
→ Left with: $\frac{2x - 7}{x + 3}$
→ Final: $\boxed{\frac{2x - 7}{x + 3}}$
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30. $\frac{8x^2 - 14x + 6}{4x - 3}$
→ Try factoring numerator: $8x^2 - 14x + 6$
Divide by 2: $2(4x^2 - 7x + 3)$
Factor inside: $4x^2 - 7x + 3 = (4x - 3)(x - 1)$? Check: 4x*x=4x², 4x*(-1)= -4x, -3*x= -3x, total -7x; (-3)*(-1)=3 ✔️
So numerator: $2(4x - 3)(x - 1)$
Denominator: $4x - 3$
→ So: $\frac{2(4x - 3)(x - 1)}{4x - 3}$ → cancel $(4x - 3)$
→ Left with: $2(x - 1)$
→ Final: $\boxed{2(x - 1)}$
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Final Answer:
1. $\frac{2}{3x^2}$
2. $\frac{3x}{4y^2}$
3. $\frac{1}{(x+1)^2}$
4. $(3x - 2)^2$
5. $-1$
6. $\frac{3}{4y}$
7. $\frac{1}{3y}$
8. $\frac{4x^2}{3}$
9. $\frac{x^2}{2}$
10. $-\frac{2x^2}{3}$
11. $\frac{a}{c}$
12. $\frac{x}{3}$
13. $-\frac{2}{x}$
14. $\frac{x + 5}{x - 7}$
15. $\frac{x - 3}{x - 6}$
16. $\frac{x - 3}{x + 3}$
17. $\frac{x + 2}{x - 1}$
18. $\frac{x - 4}{x - 5}$
19. $\frac{x - 2}{x + 1}$
20. $\frac{x + 3}{x - 1}$
21. $-\frac{x - 6}{x - 4}$
22. $-\frac{y + 3}{y + 5}$
23. $\frac{21(x - 1)}{(x + 7)(x - 2)}$
24. $\frac{1}{2(x + 1)}$
25. $\frac{5x - 1}{5x + 3}$
26. $\frac{x - 3}{x + 5}$
27. $-\frac{x - 4}{x - 2}$
28. $\frac{2(x + 2)}{x + 6}$
29. $\frac{2x - 7}{x + 3}$
30. $2(x - 1)$
Parent Tip: Review the logic above to help your child master the concept of rational equations worksheet with answers.
