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Chapter 9: Problem 16

Solve equation by using the square root property. Simplify all radicals. \(x^{2}=54\)

Short Answer

Expert verified
x = 3鈭�6 or x = -3鈭�6.

Step by step solution

01

- Understanding the Square Root Property

The square root property states that if x^2 = k, then x = 鈭歬 or x = -鈭歬. In this case, our equation is x^2 = 54.
02

- Apply the Square Root Property

Applying the property to the equation, we get x = 鈭�54 or x = - 鈭�54.
03

- Simplify the Radicals

鈭�54 can be simplified by recognizing that 54 = 9 * 6. Because 鈭�9 = 3, this allows us to simplify 鈭�54 to 3鈭�6. Thus, the equation becomes x = 3鈭�6 or x = -3鈭�6.

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Key Concepts

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

Square Root Property
One powerful tool for solving quadratic equations is the square root property. This property simplifies the process when the equation is in the form of \(x^{2} = k\).Using this property means recognizing that, if \(x^2 = k\), then the solutions for \(x\) are:
  • \(x = \sqrt{k}\)
  • \(x = -\sqrt{k}\)
In simpler words, taking the square root of both sides of the equation provides two possible values for \(x\). Understanding this concept helps in easily breaking down and solving quadratic equations.
Simplifying Radicals
Simplifying radicals is another crucial step when solving how to express your solutions clearly. A radical simplified form removes any perfect squares from inside the square root.For example, when we have \(\sqrt{54}\), breaking 54 down gives us 9 * 6. Recognizing that 9 is a perfect square (since \(3^2 = 9\)), we can simplify further:* \(\sqrt{54} = \sqrt{9 \cdot 6}\)* \(\sqrt{9} = 3\)* Hence, \(\sqrt{54} = 3 \sqrt{6}\)So, our answer for \( x^{2} = 54 \) can be written as \(x = 3 \sqrt{6}\) or \(x = -3 \sqrt{6}\). Simplifying radicals like this makes it easier to handle real numbers.
Quadratic Equations
Quadratic equations are polynomial equations of the form \(ax^{2} + bx + c = 0\). When solving these equations, several methods like factoring, using the quadratic formula, and taking square roots as discussed here can be applied.In the equation \(x^{2} = 54\), no linear term (bx) is present, simplifying our approach. Directly applying the square root property efficiently solves it. Here's a quick summary:
  • Express the equation in the standard form \(ax^{2} = k\).
  • Apply the square root to both sides:\( x = \pm \sqrt{k} \).
  • Simplify the radicals if possible.
This method is a straightforward way to handle quadratic equations without needing more complex operations.

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Most popular questions from this chapter

We can use a graphing calculator to illustrate how the graph of \(y=x^{2}\) can be transformed through arithmetic operations. In the standard viewing window of your calculator, graph the following one at a time, leaving the previous graphs on the screen as you move along. $$ Y_{1}=x^{2} \quad Y_{2}=x^{2}+3 \quad Y_{3}=x^{2}-6 $$ Describe the effect that adding or subtracting a constant has on the parabola.

Concept Check Which step is an appropriate way to begin solving the quadratic cquation \(2 x^{2}-4 x=9\) by completing the square? A. Add 4 to each side of the equation. B. Factor the left side as \(2 x(x-2)\) C. Factor the left side as \(x(2 x-4)\) D. Divide each side by 2

The area \(\mathscr{A}\) of a circle with radius \(r\) is given by the formula $$\mathscr{A}=\pi r^{2}$$ If a circle has area \(81 \pi\) in. , what is its radius?

Solve equation by using the square root property. Simplify all radicals. \(2 t^{2}+7=61\)

Solve each problem. An astronaut on the moon throws a baseball upward. The altitude (height) \(h\) of the ball, in feet, \(x\) seconds after he throws it, is given by the equation $$ h=-2.7 x^{2}+30 x+6.5 $$ At what times is the ball 12 ft above the moon's surface?
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