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Chapter 3: Problem 12

Solve. $$x-\frac{12}{x}=1$$

Short Answer

Expert verified
The solutions are x = 4 and x = -3.

Step by step solution

01

- Multiply by x

To eliminate the fraction, multiply both sides of the equation by x, giving: \[x^2 - 12 = x\]
02

- Rearrange the Equation

Move all terms to one side to set the equation to zero: \[x^2 - x - 12 = 0\]
03

- Factor the Quadratic

Factorize the quadratic equation: \[(x - 4)(x + 3) = 0\]
04

- Solve for x

Set each factor to zero and solve for x: \[x - 4 = 0 \implies x = 4\] \[x + 3 = 0 \implies x = -3\]
05

- Verify Solutions

Plug each solution back into the original equation to verify: For \(x = 4\): \[4 - \frac{12}{4} = 4 - 3 = 1 \] (True)For \(x = -3\): \[-3 - \frac{12}{-3} = -3 + 4 = 1 \] (True)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring quadratics
Factoring quadratics is a key step in solving quadratic equations. A quadratic equation has the form:
  • \(ax^2 + bx + c = 0\)
Our goal is to break this down into a product of simpler expressions, like
  • \((x - p)(x - q) = 0\)
For instance, given the quadratic equation \[x^2 - x - 12 = 0\]we look for two numbers that multiply to
  • \(-12\)
and add up to
  • \(-1\)
These numbers are
  • \(-4\) and \(3\)
So, we can factor the equation as
  • \((x - 4)(x + 3) = 0\)
This factorization allows us to find the solutions to the quadratic equation.
eliminating fractions
Eliminating fractions simplifies equations and helps us solve them more easily. When you see a fraction, your first step should be to get rid of it. In our given problem \[x - \frac{12}{x} = 1\]we can eliminate the fraction by multiplying every term by the variable in the denominator, which is
  • \(x\)
This gives: \[x(x) - x\left( \frac{12}{x} \right) = 1(x)\]Simplifying it to:
  • \(x^2 - 12 = x\)
Now the equation is easier to handle without fractions. With all terms now having whole number coefficients, we can move on to rearrange the equation for further steps.
verifying solutions
Verifying solutions ensures that our results are correct. After solving for the variable, we substitute each solution back into the original equation. In our case, the solutions are
  • \(x = 4\)
  • \(x = -3\)
For
  • \(x = 4\)
Substitute back into \[x - \frac{12}{x} = 1\]Which gives: \[4 - \frac{12}{4} = 1\]\[4 - 3 = 1\]This is true. For
  • \(x = -3\)
Substitute back into \[x - \frac{12}{x} = 1\]Which gives: \[-3 - \frac{12}{-3} = 1\]\[-3 + 4 = 1\]This is also true. By verifying both solutions, we confirm that they are correct.

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