Question

question
solve for all values of x:
$x^2(2x - 7) - x(2x - 7) - 56(2x - 7) = 0$
answer
attempt 1 out of 2
additional solution no solution
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Explanation:

Step1: Factor out common term

Notice that $(2x-7)$ is a common factor in all three terms, so we factor it out:
$$(2x - 7)(x^2 - x - 56) = 0$$

Step2: Factor quadratic expression

Factor the quadratic $x^2 - x - 56$ by finding two numbers that multiply to $-56$ and add to $-1$ (these numbers are $-8$ and $7$):
$$(2x - 7)(x - 8)(x + 7) = 0$$

Step3: Set each factor to 0

Apply the zero product property, set each factor equal to 0:

1. $2x - 7 = 0$
2. $x - 8 = 0$
3. $x + 7 = 0$

Step4: Solve for x in each equation

Solve each linear equation:

1. $2x = 7 \implies x = \frac{7}{2}$
2. $x = 8$
3. $x = -7$

Answer:

$x = -7$, $x = \frac{7}{2}$, $x = 8$