Problem 43 - Elementary Algebra

Question

Solve using the Square Root Property. \(u^{2}-14 u+49=72\)

Step-by-Step Solution

Answer

The solutions are \(u = 7 + \sqrt{95}\) and \(u = 7 - \sqrt{95}\).

Step 1: Simplify the equation

Start by moving all terms to one side of the equation to set it equal to zero. Subtract 72 from both sides: \[u^{2} - 14u + 49 - 72 = 0\] which simplifies to: \[u^{2} - 14u - 23 = 0\]

Step 2: Identify Perfect Square

Check to see if the left-hand side of the equation can be written as a perfect square. The form is already close to a perfect square trinomial. Notice that: \[(u-7)^{2} = u^{2} - 14u + 49\] Thus we rewrite the equation as: \[(u-7)^{2} = 72 + 23\] Which simplifies to: \[(u-7)^{2} = 95\]

Step 3: Apply Square Root Property

Apply the Square Root Property which states that if \((u-7)^{2} = 95\), then \(u-7 = \pm \sqrt{95}\). So: \[u - 7 = \pm \sqrt{95}\]

Step 4: Solve for u

Finally, solve for \(u\) by isolating the variable: \[u = 7 \pm \sqrt{95}\] So the solutions are: \[u = 7 + \sqrt{95}\] and \[u = 7 - \sqrt{95}\]

Key Concepts

Solving Equations

To solve equations effectively, especially quadratics, we need a reliable roadmap. Here's the step-by-step process based on the exercise example: 1. **Simplify the equation:** Move all terms to one side of the equation to set it equal to zero. This makes it easier to identify and factor the quadratic. For our example, subtract 72 from both sides: ewlineewlineu^2 - 14u + 49 - 72 = 0 simplifying to u^2 - 14u - 23 = 0. 2. **Identify Perfect Square:** Check if the quadratic term is a perfect square trinomial by matching it to the pattern ewline a^2 ± 2ab + b^2: In this case, u - 7)^2 = u^2 - 14u + 49. 3. **Apply Square Root Property:** This property states u - k)^2 = d implies u - k = ±√d. For our example, rewrite it to u - 7)^2 = 95, which gives u - 7 = ±√95. 4. **Solve for the Variable:** Isolate the variable to find the solutions. Finally, solve for u: u = 7 ±√95.