Question

Simplify the expression and find the restrictions on the variable: 3x² - 39x + 90/x² - 3x -70

Answer 1

The simplified expression is
`\((3(x - 3))/(x + 7)\)`, and the restriction is
`\(x \\eq -7\)`.

To simplify the expression
`\((3x^2 - 39x + 90)/(x^2 - 3x - 70)\)` and find the restrictions on the variable, follow these steps:

Step 1: Factor both the numerator and denominator.

Factor the numerator
`(\(3x^2 - 39x + 90\))`:

`\(3x^2 - 39x + 90 = 3(x^2 - 13x + 30)\)`

Now, factor the denominator
`(\(x^2 - 3x - 70\))`:

`\(x^2 - 3x - 70 = (x - 10)(x + 7)\)`

Step 2: Cancel out common factors.

Now, we can cancel out common factors in the fraction:

`\((3(x^2 - 13x + 30))/((x - 10)(x + 7))\)`

Step 3: Simplify further.

We can simplify the expression by factoring the quadratic inside the numerator:

`\(x^2 - 13x + 30 = (x - 10)(x - 3)\)`

So, the expression becomes:

`\((3(x - 10)(x - 3))/((x - 10)(x + 7))\)`

Step 4: Cancel out common factors, if any.

Now, we can see that there is a common factor of
`\((x - 10)\)` in both the numerator and denominator. Canceling it out:

`\((3(x - 3))/(x + 7)\)`

Step 5: Find the restrictions on the variable.

To find the restrictions on the variable, we need to consider the values of
`\(x\)` that would make the denominator equal to zero, as division by zero is undefined.

`\(x + 7 = 0\)`

Solving for
`\(x\)`:

`\(x = -7\)`

So, the restriction on the variable
`\(x\)` is
`\(x \\eq -7\)`.