Questions: Simplify the following expression completely. [ frac10 x-70x^2+3 x-70 ]

$Simplify the following expression completely. [ frac10 x-70x^2+3 x-70 ]$

Transcript text: Simplify the following expression completely. \[ \frac{10 x-70}{x^{2}+3 x-70} \] Enter the numerator and denominator separately in the boxes below. If the denominator is 1 , enter the number 1. Do not leave either box blank. Answer: $\square$ $\square$ Numerator preview: Denominator preview:

Solution

Solution Steps

To simplify the given expression, we need to factor both the numerator and the denominator. The numerator $10x - 70$ can be factored by taking out the greatest common factor. The denominator $x^2 + 3x - 70$ is a quadratic expression that can be factored by finding two numbers that multiply to $-70$ and add to $3$. Once both are factored, we can simplify the expression by canceling out common factors.

Step 1: Factor the Numerator

The numerator $10x - 70$ can be factored by taking out the greatest common factor, which is $10$: \[ 10x - 70 = 10(x - 7) \]

Step 2: Factor the Denominator

The denominator $x^2 + 3x - 70$ is a quadratic expression. We can factor it by finding two numbers that multiply to $-70$ and add to $3$. The factors are $-7$ and $10$: \[ x^2 + 3x - 70 = (x - 7)(x + 10) \]

Step 3: Simplify the Expression

Now that we have factored both the numerator and the denominator, we can write the expression as: \[ \frac{10(x - 7)}{(x - 7)(x + 10)} \] We can cancel the common factor $(x - 7)$ from the numerator and the denominator (assuming $x \neq 7$): \[ \frac{10}{x + 10} \]

Final Answer

The simplified expression is: \[ \boxed{\frac{10}{x + 10}} \]

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