Math  /  Algebra

QuestionSolve the following equation. x4214x=1\frac{x}{4}-\frac{21}{4 x}=-1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \square \}. (Type an integer or a fraction. Use a comma to separate answers as needed.) B. The solution set is {xx\{x \mid x is a real number, xx \neq \square \}. (Type an integer or a fraction. Use a comma to separate answers as needed.) C. The solution set is \varnothing.

Studdy Solution

STEP 1

What is this asking? We're asked to solve a funky equation with xx in the numerator *and* denominator, which means finding the values of xx that make the equation true! Watch out! We need to be careful not to divide by zero.
If any of our steps involve dividing by xx, we must make sure that xx isn't zero.

STEP 2

1. Rewrite the equation
2. Multiply to get rid of fractions
3. Rearrange into a standard form
4. Factor the quadratic
5. Find the solutions

STEP 3

Let's rewrite the equation to make it a bit easier to work with.
We have x4214x=1 \frac{x}{4} - \frac{21}{4x} = -1

STEP 4

Fractions can be a bit messy, so let's multiply *both* sides of the equation by 4x4x to get rid of them.
Remember, this is okay as long as xx isn't zero!
We'll check that later. 4x(x4214x)=4x(1) 4x \cdot \left( \frac{x}{4} - \frac{21}{4x} \right) = 4x \cdot (-1)

STEP 5

Distribute the 4x4x on the left side: 4xx44x214x=4x 4x \cdot \frac{x}{4} - 4x \cdot \frac{21}{4x} = -4x 4xx44x214x=4x \frac{4x \cdot x}{4} - \frac{4x \cdot 21}{4x} = -4x

STEP 6

Now, divide to one to simplify: 44xx4x4x21=4x \frac{4}{4} \cdot x \cdot x - \frac{4x}{4x} \cdot 21 = -4x x221=4x x^2 - 21 = -4x

STEP 7

Let's rearrange the equation into a standard quadratic form by adding 4x4x to both sides: x2+4x21=4x+4x x^2 + 4x - 21 = -4x + 4x x2+4x21=0 x^2 + 4x - 21 = 0

STEP 8

Now, we need to factor our quadratic equation.
We're looking for two numbers that multiply to 21-21 and add up to 44.
Those magic numbers are 77 and 3-3! (x+7)(x3)=0 (x + 7)(x - 3) = 0

STEP 9

If the product of two things is zero, then at least one of them *must* be zero.
So, either x+7=0x + 7 = 0 or x3=0x - 3 = 0.

STEP 10

If x+7=0x + 7 = 0, then x=7x = -7. If x3=0x - 3 = 0, then x=3x = 3.

STEP 11

Remember when we multiplied by 4x4x earlier?
We had to assume xx wasn't zero.
Since our solutions aren't zero, we're all good!

STEP 12

The solution set is {7,3}\{-7, 3\}.
So the answer is A, with 7,3-7, 3 in the box.

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