Factorise completely: 3.1 x2 - x − 12, 3.2 2p2 - 60p − 128, 3.2 (x − 1)2 - 2(x − 1) − 8. Simplify completely…

Question

Factorise completely: 3.1 x2 - x − 12, 3.2 2p2 - 60p − 128, 3.2 (x − 1)2 - 2(x − 1) − 8. Simplify completely: 4.1 (4x2 + 8x) / 2x.

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here are the factorisations and simplifications:

Factorise completely.

3.1 $x^2 - x - 12$

Find two numbers that multiply to $-12$ and add to $-1$ .
These numbers are 3 and $-4$ .
$x^2 - x - 12 = (x + 3)(x - 4)$
The factorised expression is $\boxed{(x + 3)(x - 4)}$ .

3.2 $2p^2 - 60p - 128$
Factor out the greatest common factor (GCF), which is 2.
$2p^2 - 60p - 128 = 2(p^2 - 30p - 64)$
Factor the quadratic expression $p^2 - 30p - 64$ .
Find two numbers that multiply to $-64$ and add to $-30$ . These numbers are 2 and $-32$ .
$= 2(p + 2)(p - 32)$
The factorised expression is $\boxed{2(p + 2)(p - 32)}$ .

3.2 $(x - 1)^2 - 2(x - 1) - 8$
Let $y = (x - 1)$ .
Substitute $y$ into the expression.
$y^2 - 2y - 8$
Factor the quadratic expression.
Find two numbers that multiply to $-8$ and add to $-2$ . These numbers are 2 and $-4$ .
$(y + 2)(y - 4)$
Substitute back $y = (x - 1)$ .
$((x - 1) + 2)((x - 1) - 4)$ $= (x + 1)(x - 5)$
The factorised expression is $\boxed{(x + 1)(x - 5)}$ .

QUESTION 4: Simplify completely.

4.1 $\frac{4x^2 + 8x}{2x}$
Factor the numerator by taking out the common factor $4x$.
$\frac{4x(x + 2)}{2x}$
Cancel out the common term $2x$ from the numerator and the denominator.
$= \frac{2(x + 2)}{1}$ $= 2(x + 2)$
The simplified expression is $\boxed{2(x + 2)}$ .

4.2 $\frac{x^2 - 2x - 15}{x^2 - 25}$
Factor the numerator $x^2 - 2x - 15$ .
Find two numbers that multiply to $-15$ and add to $-2$ . These are 3 and $-5$ .
$x^2 - 2x - 15 = (x + 3)(x - 5)$
Factor the denominator $x^2 - 25$ .
This is a difference of squares, $a^2 - b^2 = (a - b)(a + b)$ .
$x^2 - 25 = (x - 5)(x + 5)$
Substitute the factored expressions back into the fraction and simplify.
$\frac{(x + 3)(x - 5)}{(x - 5)(x + 5)}$ $= \frac{x + 3}{x + 5}$
The simplified expression is $\boxed{\frac{x + 3}{x + 5}}$ .

4.3 $\frac{x(x-1) - (1-x)}{5x^2 - 5}$
Simplify the numerator.
Note that $(1 - x) = -(x - 1)$ .
$x(x - 1) - (-(x - 1)) = x(x - 1) + (x - 1)$
Factor out the common term $(x - 1)$ .
$= (x - 1)(x + 1)$
Factor the denominator $5x^2 - 5$ .
Factor out the GCF 5, then recognize the difference of squares.
$5x^2 - 5 = 5(x^2 - 1) = 5(x - 1)(x + 1)$
Substitute the factored expressions back into the fraction and simplify.
$\frac{(x - 1)(x + 1)}{5(x - 1)(x + 1)}$ $= \frac{1}{5}$
The simplified expression is $\boxed{\frac{1}{5}}$ .

4.4 $\frac{x^2 + 8x + 15}{3x^2 + 9x} + \frac{x^2 - 25}{6x^2 + 42x}$
Simplify the first fraction $\frac{x^2 + 8x + 15}{3x^2 + 9x}$ .
Factor the numerator: $x^2 + 8x + 15 = (x + 3)(x + 5)$ .
Factor the denominator: $3x^2 + 9x = 3x(x + 3)$ .
So, the first fraction becomes: $\frac{(x + 3)(x + 5)}{3x(x + 3)} = \frac{x + 5}{3x}$ .
Simplify the second fraction $\frac{x^2 - 25}{6x^2 + 42x}$ .
Factor the numerator: $x^2 - 25 = (x - 5)(x + 5)$ .
Factor the denominator: $6x^2 + 42x = 6x(x + 7)$ .
So, the second fraction becomes: $\frac{(x - 5)(x + 5)}{6x(x + 7)}$ .
Add the simplified fractions
$\frac{x + 5}{3x} + \frac{(x - 5)(x + 5)}{6x(x + 7)}$ .
Find the least common denominator (LCD), which is $6x(x + 7)$ .
Rewrite the first fraction with the LCD:
$\frac{(x + 5)}{3x} \cdot \frac{2(x + 7)}{2(x + 7)} = \frac{2(x + 5)(x + 7)}{6x(x + 7)}$
Combine the fractions over the common denominator.
$\frac{2(x + 5)(x + 7) + (x - 5)(x + 5)}{6x(x + 7)}$
Factor out $(x + 5)$ from the numerator.
$\frac{(x + 5)[2(x + 7) + (x - 5)]}{6x(x + 7)}$
Simplify the expression inside the square brackets.
$2(x + 7) + (x - 5) = 2x + 14 + x - 5 = 3x + 9$
Factor out 3 from $3x + 9$ : $3(x + 3)$ .
Substitute this back into the numerator and simplify the entire expression.
$\frac{(x + 5) \cdot 3(x + 3)}{6x(x + 7)} = \frac{3(x + 3)(x + 5)}{6x(x + 7)}$ $= \frac{(x + 3)(x + 5)}{2x(x + 7)}$
The simplified expression is $\boxed{\frac{(x

Factorise completely: 3.1 x2 - x - 12, 3.2 2p2 - 60p - 128, 3.2 (x - 1)2 - 2(x - 1) - 8. Simplify completely: 4.1 (4x2 + 8x) / 2x.

Step-by-step solution

Find two numbers that multiply to $-12$ and add to $-1$ .

Factor out the greatest common factor (GCF), which is 2.

Factor the quadratic expression $p^2 - 30p - 64$ .

Let $y = (x - 1)$ .

Factor the quadratic expression.

Substitute back $y = (x - 1)$ .

Factor the numerator by taking out the common factor $4x$.

Cancel out the common term $2x$ from the numerator and the denominator.

Factor the numerator $x^2 - 2x - 15$ .

Factor the denominator $x^2 - 25$ .

Substitute the factored expressions back into the fraction and simplify.

Simplify the numerator.

Factor the denominator $5x^2 - 5$ .

Substitute the factored expressions back into the fraction and simplify.

Simplify the first fraction $\frac{x^2 + 8x + 15}{3x^2 + 9x}$ .

Simplify the second fraction $\frac{x^2 - 25}{6x^2 + 42x}$ .

Add the simplified fractions

Combine the fractions over the common denominator.

Factor out $(x + 5)$ from the numerator.

Simplify the expression inside the square brackets.

Substitute this back into the numerator and simplify the entire expression.

Still stuck on this one?

More Mathematics Questions

Factorise completely: 3.1 x2 - x - 12, 3.2 2p2 - 60p - 128, 3.2 (x - 1)2 - 2(x - 1) - 8. Simplify completely: 4.1 (4x2 + 8x) / 2x.

Step-by-step solution

Find two numbers that multiply to −12-12−12 and add to −1-1−1.

Factor out the greatest common factor (GCF), which is 2.

Factor the quadratic expression p2−30p−64p^2 - 30p - 64p2−30p−64.

Let y=(x−1)y = (x - 1)y=(x−1).

Factor the quadratic expression.

Substitute back y=(x−1)y = (x - 1)y=(x−1).

Factor the numerator by taking out the common factor $4x$.

Cancel out the common term $2x$ from the numerator and the denominator.

Factor the numerator x2−2x−15x^2 - 2x - 15x2−2x−15.

Factor the denominator x2−25x^2 - 25x2−25.

Substitute the factored expressions back into the fraction and simplify.

Simplify the numerator.

Factor the denominator 5x2−55x^2 - 55x2−5.

Substitute the factored expressions back into the fraction and simplify.

Simplify the first fraction x2+8x+153x2+9x\frac{x^2 + 8x + 15}{3x^2 + 9x}3x2+9xx2+8x+15​.

Simplify the second fraction x2−256x2+42x\frac{x^2 - 25}{6x^2 + 42x}6x2+42xx2−25​.

Add the simplified fractions

Combine the fractions over the common denominator.

Factor out (x+5)(x + 5)(x+5) from the numerator.

Simplify the expression inside the square brackets.

Substitute this back into the numerator and simplify the entire expression.

Still stuck on this one?

More Mathematics Questions

Find two numbers that multiply to $-12$ and add to $-1$ .

Factor the quadratic expression $p^2 - 30p - 64$ .

Let $y = (x - 1)$ .

Substitute back $y = (x - 1)$ .

Factor the numerator $x^2 - 2x - 15$ .

Factor the denominator $x^2 - 25$ .

Factor the denominator $5x^2 - 5$ .

Simplify the first fraction $\frac{x^2 + 8x + 15}{3x^2 + 9x}$ .

Simplify the second fraction $\frac{x^2 - 25}{6x^2 + 42x}$ .

Factor out $(x + 5)$ from the numerator.