Questions: Solve for u. 12 - 24/u = u + 22

Transcript text: Solve for $u$. \[ 12-\frac{24}{u}=u+22 \]

Solution

Solution Steps

To solve for $ u $, we need to isolate $ u $ on one side of the equation. First, we can eliminate the fraction by multiplying both sides by $ u $. Then, we rearrange the equation to form a quadratic equation and solve for $ u $ using the quadratic formula.

Step 1: Rearranging the Equation

We start with the equation: \[ 12 - \frac{24}{u} = u + 22 \] To eliminate the fraction, we multiply both sides by $ u $: \[ u(12) - 24 = u^2 + 22u \] This simplifies to: \[ 12u - 24 = u^2 + 22u \]

Step 2: Forming the Quadratic Equation

Rearranging the equation gives us: \[ 0 = u^2 + 22u - 12u + 24 \] This simplifies to: \[ 0 = u^2 + 10u + 24 \]

Step 3: Solving the Quadratic Equation

We can solve the quadratic equation $ u^2 + 10u + 24 = 0 $ using the quadratic formula: \[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where $ a = 1 $, $ b = 10 $, and $ c = 24 $. Calculating the discriminant: \[ b^2 - 4ac = 10^2 - 4 \cdot 1 \cdot 24 = 100 - 96 = 4 \] Thus, we have: \[ u = \frac{-10 \pm \sqrt{4}}{2 \cdot 1} = \frac{-10 \pm 2}{2} \] This results in two solutions: \[ u = \frac{-10 + 2}{2} = -4 \quad \text{and} \quad u = \frac{-10 - 2}{2} = -6 \]

Final Answer

The solutions for $ u $ are: \[ \boxed{u = -4} \quad \text{and} \quad \boxed{u = -6} \]

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