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Step-by-step solution for: Simplifying algebraic fractions worksheet no 5 (with detailed ...
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Step-by-step solution for: Simplifying algebraic fractions worksheet no 5 (with detailed ...
To solve the given problems involving simplifying algebraic fractions by factorizing, we will follow these steps:
1. Factorize the numerator and denominator of each fraction.
2. Cancel out any common factors between the numerator and denominator.
3. Write the simplified expression.
Let's go through each problem step by step.
---
#### A1. Simplify \(\frac{x+3}{x^2 + x - 6}\)
- Step 1: Factorize the denominator \(x^2 + x - 6\).
\[
x^2 + x - 6 = (x + 3)(x - 2)
\]
- Step 2: Rewrite the fraction.
\[
\frac{x+3}{x^2 + x - 6} = \frac{x+3}{(x+3)(x-2)}
\]
- Step 3: Cancel the common factor \(x+3\).
\[
\frac{x+3}{(x+3)(x-2)} = \frac{1}{x-2}
\]
- Final Answer:
\[
\boxed{\frac{1}{x-2}}
\]
---
#### A2. Simplify \(\frac{x+6}{x^2 + 10x + 24}\)
- Step 1: Factorize the denominator \(x^2 + 10x + 24\).
\[
x^2 + 10x + 24 = (x + 6)(x + 4)
\]
- Step 2: Rewrite the fraction.
\[
\frac{x+6}{x^2 + 10x + 24} = \frac{x+6}{(x+6)(x+4)}
\]
- Step 3: Cancel the common factor \(x+6\).
\[
\frac{x+6}{(x+6)(x+4)} = \frac{1}{x+4}
\]
- Final Answer:
\[
\boxed{\frac{1}{x+4}}
\]
---
#### A3. Simplify \(\frac{2x-18}{x^2 - 12x + 27}\)
- Step 1: Factorize the numerator \(2x - 18\).
\[
2x - 18 = 2(x - 9)
\]
- Step 2: Factorize the denominator \(x^2 - 12x + 27\).
\[
x^2 - 12x + 27 = (x - 9)(x - 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{2x-18}{x^2 - 12x + 27} = \frac{2(x-9)}{(x-9)(x-3)}
\]
- Step 4: Cancel the common factor \(x-9\).
\[
\frac{2(x-9)}{(x-9)(x-3)} = \frac{2}{x-3}
\]
- Final Answer:
\[
\boxed{\frac{2}{x-3}}
\]
---
#### A4. Simplify \(\frac{x^2 - 7x}{x^2 - 2x - 35}\)
- Step 1: Factorize the numerator \(x^2 - 7x\).
\[
x^2 - 7x = x(x - 7)
\]
- Step 2: Factorize the denominator \(x^2 - 2x - 35\).
\[
x^2 - 2x - 35 = (x - 7)(x + 5)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 7x}{x^2 - 2x - 35} = \frac{x(x-7)}{(x-7)(x+5)}
\]
- Step 4: Cancel the common factor \(x-7\).
\[
\frac{x(x-7)}{(x-7)(x+5)} = \frac{x}{x+5}
\]
- Final Answer:
\[
\boxed{\frac{x}{x+5}}
\]
---
#### B1. Simplify \(\frac{x^2 - 49}{x^2 + 2x - 35}\)
- Step 1: Factorize the numerator \(x^2 - 49\).
\[
x^2 - 49 = (x - 7)(x + 7)
\]
- Step 2: Factorize the denominator \(x^2 + 2x - 35\).
\[
x^2 + 2x - 35 = (x - 5)(x + 7)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 49}{x^2 + 2x - 35} = \frac{(x-7)(x+7)}{(x-5)(x+7)}
\]
- Step 4: Cancel the common factor \(x+7\).
\[
\frac{(x-7)(x+7)}{(x-5)(x+7)} = \frac{x-7}{x-5}
\]
- Final Answer:
\[
\boxed{\frac{x-7}{x-5}}
\]
---
#### B2. Simplify \(\frac{x^2 - 4}{x^2 + 6x - 16}\)
- Step 1: Factorize the numerator \(x^2 - 4\).
\[
x^2 - 4 = (x - 2)(x + 2)
\]
- Step 2: Factorize the denominator \(x^2 + 6x - 16\).
\[
x^2 + 6x - 16 = (x + 8)(x - 2)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 4}{x^2 + 6x - 16} = \frac{(x-2)(x+2)}{(x+8)(x-2)}
\]
- Step 4: Cancel the common factor \(x-2\).
\[
\frac{(x-2)(x+2)}{(x+8)(x-2)} = \frac{x+2}{x+8}
\]
- Final Answer:
\[
\boxed{\frac{x+2}{x+8}}
\]
---
#### B3. Simplify \(\frac{x^2 - 16}{x^2 + 2x - 24}\)
- Step 1: Factorize the numerator \(x^2 - 16\).
\[
x^2 - 16 = (x - 4)(x + 4)
\]
- Step 2: Factorize the denominator \(x^2 + 2x - 24\).
\[
x^2 + 2x - 24 = (x + 6)(x - 4)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 16}{x^2 + 2x - 24} = \frac{(x-4)(x+4)}{(x+6)(x-4)}
\]
- Step 4: Cancel the common factor \(x-4\).
\[
\frac{(x-4)(x+4)}{(x+6)(x-4)} = \frac{x+4}{x+6}
\]
- Final Answer:
\[
\boxed{\frac{x+4}{x+6}}
\]
---
#### B4. Simplify \(\frac{x^2 - 36}{x^2 + 3x - 18}\)
- Step 1: Factorize the numerator \(x^2 - 36\).
\[
x^2 - 36 = (x - 6)(x + 6)
\]
- Step 2: Factorize the denominator \(x^2 + 3x - 18\).
\[
x^2 + 3x - 18 = (x + 6)(x - 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 36}{x^2 + 3x - 18} = \frac{(x-6)(x+6)}{(x+6)(x-3)}
\]
- Step 4: Cancel the common factor \(x+6\).
\[
\frac{(x-6)(x+6)}{(x+6)(x-3)} = \frac{x-6}{x-3}
\]
- Final Answer:
\[
\boxed{\frac{x-6}{x-3}}
\]
---
#### C1. Simplify \(\frac{x^2 + 6x - 16}{2x^2 - x - 6}\)
- Step 1: Factorize the numerator \(x^2 + 6x - 16\).
\[
x^2 + 6x - 16 = (x + 8)(x - 2)
\]
- Step 2: Factorize the denominator \(2x^2 - x - 6\).
\[
2x^2 - x - 6 = (2x + 3)(x - 2)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 + 6x - 16}{2x^2 - x - 6} = \frac{(x+8)(x-2)}{(2x+3)(x-2)}
\]
- Step 4: Cancel the common factor \(x-2\).
\[
\frac{(x+8)(x-2)}{(2x+3)(x-2)} = \frac{x+8}{2x+3}
\]
- Final Answer:
\[
\boxed{\frac{x+8}{2x+3}}
\]
---
#### C2. Simplify \(\frac{x^2 - 7x + 10}{3x^2 - 5x - 2}\)
- Step 1: Factorize the numerator \(x^2 - 7x + 10\).
\[
x^2 - 7x + 10 = (x - 5)(x - 2)
\]
- Step 2: Factorize the denominator \(3x^2 - 5x - 2\).
\[
3x^2 - 5x - 2 = (3x + 1)(x - 2)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 7x + 10}{3x^2 - 5x - 2} = \frac{(x-5)(x-2)}{(3x+1)(x-2)}
\]
- Step 4: Cancel the common factor \(x-2\).
\[
\frac{(x-5)(x-2)}{(3x+1)(x-2)} = \frac{x-5}{3x+1}
\]
- Final Answer:
\[
\boxed{\frac{x-5}{3x+1}}
\]
---
#### C3. Simplify \(\frac{x^2 + 3x}{5x^2 + 19x + 12}\)
- Step 1: Factorize the numerator \(x^2 + 3x\).
\[
x^2 + 3x = x(x + 3)
\]
- Step 2: Factorize the denominator \(5x^2 + 19x + 12\).
\[
5x^2 + 19x + 12 = (5x + 4)(x + 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 + 3x}{5x^2 + 19x + 12} = \frac{x(x+3)}{(5x+4)(x+3)}
\]
- Step 4: Cancel the common factor \(x+3\).
\[
\frac{x(x+3)}{(5x+4)(x+3)} = \frac{x}{5x+4}
\]
- Final Answer:
\[
\boxed{\frac{x}{5x+4}}
\]
---
#### C4. Simplify \(\frac{x^2 + 4x - 21}{4x^2 - 17x + 15}\)
- Step 1: Factorize the numerator \(x^2 + 4x - 21\).
\[
x^2 + 4x - 21 = (x + 7)(x - 3)
\]
- Step 2: Factorize the denominator \(4x^2 - 17x + 15\).
\[
4x^2 - 17x + 15 = (4x - 5)(x - 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 + 4x - 21}{4x^2 - 17x + 15} = \frac{(x+7)(x-3)}{(4x-5)(x-3)}
\]
- Step 4: Cancel the common factor \(x-3\).
\[
\frac{(x+7)(x-3)}{(4x-5)(x-3)} = \frac{x+7}{4x-5}
\]
- Final Answer:
\[
\boxed{\frac{x+7}{4x-5}}
\]
---
#### D1. Simplify \(\frac{2x^2 - 5x - 12}{2x^2 - 11x + 12}\)
- Step 1: Factorize the numerator \(2x^2 - 5x - 12\).
\[
2x^2 - 5x - 12 = (2x + 3)(x - 4)
\]
- Step 2: Factorize the denominator \(2x^2 - 11x + 12\).
\[
2x^2 - 11x + 12 = (2x - 3)(x - 4)
\]
- Step 3: Rewrite the fraction.
\[
\frac{2x^2 - 5x - 12}{2x^2 - 11x + 12} = \frac{(2x+3)(x-4)}{(2x-3)(x-4)}
\]
- Step 4: Cancel the common factor \(x-4\).
\[
\frac{(2x+3)(x-4)}{(2x-3)(x-4)} = \frac{2x+3}{2x-3}
\]
- Final Answer:
\[
\boxed{\frac{2x+3}{2x-3}}
\]
---
#### D2. Simplify \(\frac{5x^2 - 24x - 5}{5x^2 - 14x - 3}\)
- Step 1: Factorize the numerator \(5x^2 - 24x - 5\).
\[
5x^2 - 24x - 5 = (5x + 1)(x - 5)
\]
- Step 2: Factorize the denominator \(5x^2 - 14x - 3\).
\[
5x^2 - 14x - 3 = (5x + 1)(x - 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{5x^2 - 24x - 5}{5x^2 - 14x - 3} = \frac{(5x+1)(x-5)}{(5x+1)(x-3)}
\]
- Step 4: Cancel the common factor \(5x+1\).
\[
\frac{(5x+1)(x-5)}{(5x+1)(x-3)} = \frac{x-5}{x-3}
\]
- Final Answer:
\[
\boxed{\frac{x-5}{x-3}}
\]
---
#### D3. Simplify \(\frac{x^2 - 16}{2x^2 + 13x + 20}\)
- Step 1: Factorize the numerator \(x^2 - 16\).
\[
x^2 - 16 = (x - 4)(x + 4)
\]
- Step 2: Factorize the denominator \(2x^2 + 13x + 20\).
\[
2x^2 + 13x + 20 = (2x + 5)(x + 4)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 16}{2x^2 + 13x + 20} = \frac{(x-4)(x+4)}{(2x+5)(x+4)}
\]
- Step 4: Cancel the common factor \(x+4\).
\[
\frac{(x-4)(x+4)}{(2x+5)(x+4)} = \frac{x-4}{2x+5}
\]
- Final Answer:
\[
\boxed{\frac{x-4}{2x+5}}
\]
---
#### D4. Simplify \(\frac{6x^2 - 11x + 4}{4x^2 + 8x - 5}\)
- Step 1: Factorize the numerator \(6x^2 - 11x + 4\).
\[
6x^2 - 11x + 4 = (3x - 4)(2x - 1)
\]
- Step 2: Factorize the denominator \(4x^2 + 8x - 5\).
\[
4x^2 + 8x - 5 = (2x + 5)(2x - 1)
\]
- Step 3: Rewrite the fraction.
\[
\frac{6x^2 - 11x + 4}{4x^2 + 8x - 5} = \frac{(3x-4)(2x-1)}{(2x+5)(2x-1)}
\]
- Step 4: Cancel the common factor \(2x-1\).
\[
\frac{(3x-4)(2x-1)}{(2x+5)(2x-1)} = \frac{3x-4}{2x+5}
\]
- Final Answer:
\[
\boxed{\frac{3x-4}{2x+5}}
\]
---
\[
\boxed{
\begin{array}{ll}
\text{A1: } \frac{1}{x-2} & \text{A2: } \frac{1}{x+4} \\
\text{A3: } \frac{2}{x-3} & \text{A4: } \frac{x}{x+5} \\
\text{B1: } \frac{x-7}{x-5} & \text{B2: } \frac{x+2}{x+8} \\
\text{B3: } \frac{x+4}{x+6} & \text{B4: } \frac{x-6}{x-3} \\
\text{C1: } \frac{x+8}{2x+3} & \text{C2: } \frac{x-5}{3x+1} \\
\text{C3: } \frac{x}{5x+4} & \text{C4: } \frac{x+7}{4x-5} \\
\text{D1: } \frac{2x+3}{2x-3} & \text{D2: } \frac{x-5}{x-3} \\
\text{D3: } \frac{x-4}{2x+5} & \text{D4: } \frac{3x-4}{2x+5} \\
\end{array}
}
\]
1. Factorize the numerator and denominator of each fraction.
2. Cancel out any common factors between the numerator and denominator.
3. Write the simplified expression.
Let's go through each problem step by step.
---
Section A: Simplifying by Factorizing
#### A1. Simplify \(\frac{x+3}{x^2 + x - 6}\)
- Step 1: Factorize the denominator \(x^2 + x - 6\).
\[
x^2 + x - 6 = (x + 3)(x - 2)
\]
- Step 2: Rewrite the fraction.
\[
\frac{x+3}{x^2 + x - 6} = \frac{x+3}{(x+3)(x-2)}
\]
- Step 3: Cancel the common factor \(x+3\).
\[
\frac{x+3}{(x+3)(x-2)} = \frac{1}{x-2}
\]
- Final Answer:
\[
\boxed{\frac{1}{x-2}}
\]
---
#### A2. Simplify \(\frac{x+6}{x^2 + 10x + 24}\)
- Step 1: Factorize the denominator \(x^2 + 10x + 24\).
\[
x^2 + 10x + 24 = (x + 6)(x + 4)
\]
- Step 2: Rewrite the fraction.
\[
\frac{x+6}{x^2 + 10x + 24} = \frac{x+6}{(x+6)(x+4)}
\]
- Step 3: Cancel the common factor \(x+6\).
\[
\frac{x+6}{(x+6)(x+4)} = \frac{1}{x+4}
\]
- Final Answer:
\[
\boxed{\frac{1}{x+4}}
\]
---
#### A3. Simplify \(\frac{2x-18}{x^2 - 12x + 27}\)
- Step 1: Factorize the numerator \(2x - 18\).
\[
2x - 18 = 2(x - 9)
\]
- Step 2: Factorize the denominator \(x^2 - 12x + 27\).
\[
x^2 - 12x + 27 = (x - 9)(x - 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{2x-18}{x^2 - 12x + 27} = \frac{2(x-9)}{(x-9)(x-3)}
\]
- Step 4: Cancel the common factor \(x-9\).
\[
\frac{2(x-9)}{(x-9)(x-3)} = \frac{2}{x-3}
\]
- Final Answer:
\[
\boxed{\frac{2}{x-3}}
\]
---
#### A4. Simplify \(\frac{x^2 - 7x}{x^2 - 2x - 35}\)
- Step 1: Factorize the numerator \(x^2 - 7x\).
\[
x^2 - 7x = x(x - 7)
\]
- Step 2: Factorize the denominator \(x^2 - 2x - 35\).
\[
x^2 - 2x - 35 = (x - 7)(x + 5)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 7x}{x^2 - 2x - 35} = \frac{x(x-7)}{(x-7)(x+5)}
\]
- Step 4: Cancel the common factor \(x-7\).
\[
\frac{x(x-7)}{(x-7)(x+5)} = \frac{x}{x+5}
\]
- Final Answer:
\[
\boxed{\frac{x}{x+5}}
\]
---
Section B: Simplifying by Factorizing
#### B1. Simplify \(\frac{x^2 - 49}{x^2 + 2x - 35}\)
- Step 1: Factorize the numerator \(x^2 - 49\).
\[
x^2 - 49 = (x - 7)(x + 7)
\]
- Step 2: Factorize the denominator \(x^2 + 2x - 35\).
\[
x^2 + 2x - 35 = (x - 5)(x + 7)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 49}{x^2 + 2x - 35} = \frac{(x-7)(x+7)}{(x-5)(x+7)}
\]
- Step 4: Cancel the common factor \(x+7\).
\[
\frac{(x-7)(x+7)}{(x-5)(x+7)} = \frac{x-7}{x-5}
\]
- Final Answer:
\[
\boxed{\frac{x-7}{x-5}}
\]
---
#### B2. Simplify \(\frac{x^2 - 4}{x^2 + 6x - 16}\)
- Step 1: Factorize the numerator \(x^2 - 4\).
\[
x^2 - 4 = (x - 2)(x + 2)
\]
- Step 2: Factorize the denominator \(x^2 + 6x - 16\).
\[
x^2 + 6x - 16 = (x + 8)(x - 2)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 4}{x^2 + 6x - 16} = \frac{(x-2)(x+2)}{(x+8)(x-2)}
\]
- Step 4: Cancel the common factor \(x-2\).
\[
\frac{(x-2)(x+2)}{(x+8)(x-2)} = \frac{x+2}{x+8}
\]
- Final Answer:
\[
\boxed{\frac{x+2}{x+8}}
\]
---
#### B3. Simplify \(\frac{x^2 - 16}{x^2 + 2x - 24}\)
- Step 1: Factorize the numerator \(x^2 - 16\).
\[
x^2 - 16 = (x - 4)(x + 4)
\]
- Step 2: Factorize the denominator \(x^2 + 2x - 24\).
\[
x^2 + 2x - 24 = (x + 6)(x - 4)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 16}{x^2 + 2x - 24} = \frac{(x-4)(x+4)}{(x+6)(x-4)}
\]
- Step 4: Cancel the common factor \(x-4\).
\[
\frac{(x-4)(x+4)}{(x+6)(x-4)} = \frac{x+4}{x+6}
\]
- Final Answer:
\[
\boxed{\frac{x+4}{x+6}}
\]
---
#### B4. Simplify \(\frac{x^2 - 36}{x^2 + 3x - 18}\)
- Step 1: Factorize the numerator \(x^2 - 36\).
\[
x^2 - 36 = (x - 6)(x + 6)
\]
- Step 2: Factorize the denominator \(x^2 + 3x - 18\).
\[
x^2 + 3x - 18 = (x + 6)(x - 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 36}{x^2 + 3x - 18} = \frac{(x-6)(x+6)}{(x+6)(x-3)}
\]
- Step 4: Cancel the common factor \(x+6\).
\[
\frac{(x-6)(x+6)}{(x+6)(x-3)} = \frac{x-6}{x-3}
\]
- Final Answer:
\[
\boxed{\frac{x-6}{x-3}}
\]
---
Section C: Simplifying by Factorizing
#### C1. Simplify \(\frac{x^2 + 6x - 16}{2x^2 - x - 6}\)
- Step 1: Factorize the numerator \(x^2 + 6x - 16\).
\[
x^2 + 6x - 16 = (x + 8)(x - 2)
\]
- Step 2: Factorize the denominator \(2x^2 - x - 6\).
\[
2x^2 - x - 6 = (2x + 3)(x - 2)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 + 6x - 16}{2x^2 - x - 6} = \frac{(x+8)(x-2)}{(2x+3)(x-2)}
\]
- Step 4: Cancel the common factor \(x-2\).
\[
\frac{(x+8)(x-2)}{(2x+3)(x-2)} = \frac{x+8}{2x+3}
\]
- Final Answer:
\[
\boxed{\frac{x+8}{2x+3}}
\]
---
#### C2. Simplify \(\frac{x^2 - 7x + 10}{3x^2 - 5x - 2}\)
- Step 1: Factorize the numerator \(x^2 - 7x + 10\).
\[
x^2 - 7x + 10 = (x - 5)(x - 2)
\]
- Step 2: Factorize the denominator \(3x^2 - 5x - 2\).
\[
3x^2 - 5x - 2 = (3x + 1)(x - 2)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 7x + 10}{3x^2 - 5x - 2} = \frac{(x-5)(x-2)}{(3x+1)(x-2)}
\]
- Step 4: Cancel the common factor \(x-2\).
\[
\frac{(x-5)(x-2)}{(3x+1)(x-2)} = \frac{x-5}{3x+1}
\]
- Final Answer:
\[
\boxed{\frac{x-5}{3x+1}}
\]
---
#### C3. Simplify \(\frac{x^2 + 3x}{5x^2 + 19x + 12}\)
- Step 1: Factorize the numerator \(x^2 + 3x\).
\[
x^2 + 3x = x(x + 3)
\]
- Step 2: Factorize the denominator \(5x^2 + 19x + 12\).
\[
5x^2 + 19x + 12 = (5x + 4)(x + 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 + 3x}{5x^2 + 19x + 12} = \frac{x(x+3)}{(5x+4)(x+3)}
\]
- Step 4: Cancel the common factor \(x+3\).
\[
\frac{x(x+3)}{(5x+4)(x+3)} = \frac{x}{5x+4}
\]
- Final Answer:
\[
\boxed{\frac{x}{5x+4}}
\]
---
#### C4. Simplify \(\frac{x^2 + 4x - 21}{4x^2 - 17x + 15}\)
- Step 1: Factorize the numerator \(x^2 + 4x - 21\).
\[
x^2 + 4x - 21 = (x + 7)(x - 3)
\]
- Step 2: Factorize the denominator \(4x^2 - 17x + 15\).
\[
4x^2 - 17x + 15 = (4x - 5)(x - 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 + 4x - 21}{4x^2 - 17x + 15} = \frac{(x+7)(x-3)}{(4x-5)(x-3)}
\]
- Step 4: Cancel the common factor \(x-3\).
\[
\frac{(x+7)(x-3)}{(4x-5)(x-3)} = \frac{x+7}{4x-5}
\]
- Final Answer:
\[
\boxed{\frac{x+7}{4x-5}}
\]
---
Section D: Simplifying by Factorizing
#### D1. Simplify \(\frac{2x^2 - 5x - 12}{2x^2 - 11x + 12}\)
- Step 1: Factorize the numerator \(2x^2 - 5x - 12\).
\[
2x^2 - 5x - 12 = (2x + 3)(x - 4)
\]
- Step 2: Factorize the denominator \(2x^2 - 11x + 12\).
\[
2x^2 - 11x + 12 = (2x - 3)(x - 4)
\]
- Step 3: Rewrite the fraction.
\[
\frac{2x^2 - 5x - 12}{2x^2 - 11x + 12} = \frac{(2x+3)(x-4)}{(2x-3)(x-4)}
\]
- Step 4: Cancel the common factor \(x-4\).
\[
\frac{(2x+3)(x-4)}{(2x-3)(x-4)} = \frac{2x+3}{2x-3}
\]
- Final Answer:
\[
\boxed{\frac{2x+3}{2x-3}}
\]
---
#### D2. Simplify \(\frac{5x^2 - 24x - 5}{5x^2 - 14x - 3}\)
- Step 1: Factorize the numerator \(5x^2 - 24x - 5\).
\[
5x^2 - 24x - 5 = (5x + 1)(x - 5)
\]
- Step 2: Factorize the denominator \(5x^2 - 14x - 3\).
\[
5x^2 - 14x - 3 = (5x + 1)(x - 3)
\]
- Step 3: Rewrite the fraction.
\[
\frac{5x^2 - 24x - 5}{5x^2 - 14x - 3} = \frac{(5x+1)(x-5)}{(5x+1)(x-3)}
\]
- Step 4: Cancel the common factor \(5x+1\).
\[
\frac{(5x+1)(x-5)}{(5x+1)(x-3)} = \frac{x-5}{x-3}
\]
- Final Answer:
\[
\boxed{\frac{x-5}{x-3}}
\]
---
#### D3. Simplify \(\frac{x^2 - 16}{2x^2 + 13x + 20}\)
- Step 1: Factorize the numerator \(x^2 - 16\).
\[
x^2 - 16 = (x - 4)(x + 4)
\]
- Step 2: Factorize the denominator \(2x^2 + 13x + 20\).
\[
2x^2 + 13x + 20 = (2x + 5)(x + 4)
\]
- Step 3: Rewrite the fraction.
\[
\frac{x^2 - 16}{2x^2 + 13x + 20} = \frac{(x-4)(x+4)}{(2x+5)(x+4)}
\]
- Step 4: Cancel the common factor \(x+4\).
\[
\frac{(x-4)(x+4)}{(2x+5)(x+4)} = \frac{x-4}{2x+5}
\]
- Final Answer:
\[
\boxed{\frac{x-4}{2x+5}}
\]
---
#### D4. Simplify \(\frac{6x^2 - 11x + 4}{4x^2 + 8x - 5}\)
- Step 1: Factorize the numerator \(6x^2 - 11x + 4\).
\[
6x^2 - 11x + 4 = (3x - 4)(2x - 1)
\]
- Step 2: Factorize the denominator \(4x^2 + 8x - 5\).
\[
4x^2 + 8x - 5 = (2x + 5)(2x - 1)
\]
- Step 3: Rewrite the fraction.
\[
\frac{6x^2 - 11x + 4}{4x^2 + 8x - 5} = \frac{(3x-4)(2x-1)}{(2x+5)(2x-1)}
\]
- Step 4: Cancel the common factor \(2x-1\).
\[
\frac{(3x-4)(2x-1)}{(2x+5)(2x-1)} = \frac{3x-4}{2x+5}
\]
- Final Answer:
\[
\boxed{\frac{3x-4}{2x+5}}
\]
---
Final Answers:
\[
\boxed{
\begin{array}{ll}
\text{A1: } \frac{1}{x-2} & \text{A2: } \frac{1}{x+4} \\
\text{A3: } \frac{2}{x-3} & \text{A4: } \frac{x}{x+5} \\
\text{B1: } \frac{x-7}{x-5} & \text{B2: } \frac{x+2}{x+8} \\
\text{B3: } \frac{x+4}{x+6} & \text{B4: } \frac{x-6}{x-3} \\
\text{C1: } \frac{x+8}{2x+3} & \text{C2: } \frac{x-5}{3x+1} \\
\text{C3: } \frac{x}{5x+4} & \text{C4: } \frac{x+7}{4x-5} \\
\text{D1: } \frac{2x+3}{2x-3} & \text{D2: } \frac{x-5}{x-3} \\
\text{D3: } \frac{x-4}{2x+5} & \text{D4: } \frac{3x-4}{2x+5} \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of simplifying algebraic fractions worksheet.
