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Chapter 9: Problem 59

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

Short Answer

Expert verified
u = 7 \pm 6 \sqrt{2}.

Step by step solution

01

Simplify the Equation

First, simplify the given equation: \[ u^{2} - 14u + 49 = 72 \]Move all terms to one side to set the equation equal to zero.\[ u^{2} - 14u + 49 - 72 = 0 \]Simplify the constants:\[ u^{2} - 14u - 23 = 0 \]
02

Add/Subtract to Isolate the Perfect Square

Now, re-write the quadratic equation to recognize the perfect square form. Notice that \( u^2 - 14u + 49 \) is a perfect square equation where:\[ (u - 7)^2 = 72 \].
03

Apply the Square Root Property

Using the Square Root Property, take the square root of both sides of the equation:\[ u - 7 = \pm \sqrt{72} \]Simplify \( \sqrt{72} \):\[ u - 7 = \pm 6 \sqrt{2} \]
04

Solve for u

Finally, solve for \( u \):\[ u = 7 \pm 6 \sqrt{2} \]

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Key Concepts

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

Square Root Property
The Square Root Property is a useful tool when dealing with quadratic equations, especially those that can be expressed as perfect squares. This property states that if we have an equation of the form \( (x - a)^2 = b \), then the solutions can be found by taking the square root of both sides. Specifically, \( x - a = \pm \sqrt{b} \).
In our exercise, the equation \((u - 7)^2 = 72 \) simplified to \( u - 7 = \pm 6 \sqrt{2} \) upon taking the square root of both sides.
  • This provided two potential solutions for \( u \).
  • By solving these, we get \( u = 7 + 6 \sqrt{2} \) and \( u = 7 - 6 \sqrt{2} \).
This method is direct and often simplifies the process of solving quadratic equations when applicable.
Perfect Square Trinomial
A perfect square trinomial is a quadratic expression of the form \( a^2 + 2ab + b^2 \), which can be factored into \( (a + b)^2 \). In our exercise, the given quadratic equation \( u^{2} - 14u + 49 = 72 \) can be re-written in the perfect square form \((u - 7)^2 = 72 \).
Here,
  • The middle term \( -14u \) is \( -2 \times u \times 7 \).
  • The last term \( 49 \) is \( 7^2 \).
Identifying the trinomial correctly helps in recognizing that the quadratic equation can be simplified using the Square Root Property.
The identification of these forms can make seemingly complex quadratic equations much easier to manage.
Solving Step-by-Step
Understanding how to solve quadratic equations step-by-step is crucial for mastering the material. The breakdown of the problem into smaller steps helps in grasping the fundamental principles behind each method.
In our exercise:
  • Step 1: Simplifying the given equation \( u^{2} - 14u + 49 = 72 \).
  • Step 2: Rewriting the quadratic equation \(u^{2} - 14u - 23 = 0\) to have it all terms on one side.
  • Step 3: Recognizing the perfect square form \((u - 7)^2 = 72 \).
  • Step 4: Applying the Square Root Property \( u - 7 = \pm \sqrt{72} \).
  • Step 5: Simplifying the square root, \( \sqrt{72} = 6 \sqrt{2} \).
  • Step 6: Solving for \( u \) to get solutions \(u = 7 \pm 6 \sqrt{2}\).
This structured approach ensures a clear understanding of each part of the process and enhances problem-solving skills.

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