Factor the numerator by taking out the common factor of -3.

Mathematics
Factor the numerator by taking out the common factor of -3.

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Answer

−3a−3\frac{-3}{a - 3}

Here are the factorizations and simplifications for each expression:

a) To factorize and simplify −3a+18a2−9a+18\frac{-3a + 18}{a^2 - 9a + 18}: Step 1: Factor the numerator by taking out the common factor of -3. −3a+18=−3(a−6)-3a + 18 = -3(a - 6) Step 2: Factor the denominator. We need two numbers that multiply to 18 and add to -9. These numbers are -3 and -6. a2−9a+18=(a−3)(a−6)a^2 - 9a + 18 = (a - 3)(a - 6) Step 3: Substitute the factored expressions back into the fraction and simplify by canceling the common factor (a−6)(a-6). −3(a−6)(a−3)(a−6)=−3a−3\frac{-3(a - 6)}{(a - 3)(a - 6)} = \frac{-3}{a - 3} The simplified expression is −3a−3\boxed{\frac{-3}{a - 3}}.

b) To factorize and simplify 4x2−14x2−8x\frac{4x^2 - 1}{4x^2 - 8x}: Step 1: Factor the numerator. 4x2−14x^2 - 1 is a difference of squares, (2x)2−12(2x)^2 - 1^2. 4x2−1=(2x−1)(2x+1)4x^2 - 1 = (2x - 1)(2x + 1) Step 2: Factor the denominator by taking out the common factor of 4x4x. 4x2−8x=4x(x−2)4x^2 - 8x = 4x(x - 2) Step 3: Substitute the factored expressions back into the fraction. (2x−1)(2x+1)4x(x−2)\frac{(2x - 1)(2x + 1)}{4x(x - 2)} There are no common factors to cancel. The simplified expression is (2x−1)(2x+1)4x(x−2)\boxed{\frac{(2x - 1)(2x + 1)}{4x(x - 2)}}.

c) To factorize and simplify a2−25(a−5)2×a−55+a\frac{a^2 - 25}{(a-5)^2} \times \frac{a-5}{5+a}: Step 1: Factor a2−25a^2 - 25 in the first numerator. It is a difference of squares. a2−25=(a−5)(a+5)a^2 - 25 = (a - 5)(a + 5) Step 2: Rewrite the expression with the factored term. Note that 5+a5+a is the same as a+5a+5. (a−5)(a+5)(a−5)2×a−5a+5\frac{(a - 5)(a + 5)}{(a-5)^2} \times \frac{a-5}{a+5} Step 3: Simplify by canceling common factors. (a−5)(a+5)(a−5)(a−5)×a−5a+5=1\frac{(a - 5)(a + 5)}{(a-5)(a-5)} \times \frac{a-5}{a+5} = 1 The simplified expression is 1\boxed{1}.

d) To factorize and simplify x2−1(x−1)2\frac{x^2 - 1}{(x-1)^2}: Step 1: Factor the numerator. x2−1x^2 - 1 is a difference of squares. x2−1=(x−1)(x+1)x^2 - 1 = (x - 1)(x + 1) Step 2: Rewrite the expression with the factored term. (x−1)(x+1)(x−1)2\frac{(x - 1)(x + 1)}{(x-1)^2} Step 3: Simplify by canceling one common factor of (x−1)(x-1) from the numerator and denominator. x+1x−1\frac{x + 1}{x - 1} The simplified expression is x+1x−1\boxed{\frac{x + 1}{x - 1}}.

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a) To factorize and simplify (-3a + 18)/(a^2 - 9a + 18): Step 1: Factor the numerator by taking out the common factor of -3.

Factor the numerator by taking out the common factor of -3.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Here are the factorizations and simplifications for each expression: a) To factorize and simplify (-3a + 18)/(a^2 - 9a + 18): Step 1: Factor the numerator by taking out the common factor of -3. -3a + 18 = -3(a - 6) Step 2: Factor the denominator. We need two numbers that multiply to 18 and add to -9. These numbers are -3 and -6. a^2 - 9a + 18 = (a - 3)(a - 6) Step 3: Substitute the factored expressions back into the fraction and simplify by canceling the common factor (a-6). (-3(a - 6))/((a - 3)(a - 6)) = (-3)/(a - 3) The simplified expression is (-3)/(a - 3). b) To factorize and simplify (4x^2 - 1)/(4x^2 - 8x): Step 1: Factor the numerator. 4x^2 - 1 is a difference of squares, (2x)^2 - 1^2. 4x^2 - 1 = (2x - 1)(2x + 1) Step 2: Factor the denominator by taking out the common factor of 4x. 4x^2 - 8x = 4x(x - 2) Step 3: Substitute the factored expressions back into the fraction. ((2x - 1)(2x + 1))/(4x(x - 2)) There are no common factors to cancel. The simplified expression is ((2x - 1)(2x + 1))/(4x(x - 2)). c) To factorize and simplify (a^2 - 25)/((a-5)^2) × (a-5)/(5+a): Step 1: Factor a^2 - 25 in the first numerator. It is a difference of squares. a^2 - 25 = (a - 5)(a + 5) Step 2: Rewrite the expression with the factored term. Note that 5+a is the same as a+5. ((a - 5)(a + 5))/((a-5)^2) × (a-5)/(a+5) Step 3: Simplify by canceling common factors. ((a - 5)(a + 5))/((a-5)(a-5)) × (a-5)/(a+5) = 1 The simplified expression is 1. d) To factorize and simplify (x^2 - 1)/((x-1)^2): Step 1: Factor the numerator. x^2 - 1 is a difference of squares. x^2 - 1 = (x - 1)(x + 1) Step 2: Rewrite the expression with the factored term. ((x - 1)(x + 1))/((x-1)^2) Step 3: Simplify by canceling one common factor of (x-1) from the numerator and denominator. (x + 1)/(x - 1) The simplified expression is (x + 1)/(x - 1). That's 2 down. 3 left today — send the next one.