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Chapter 9: Problem 47

Solve using the Square Root Property. $$2 r^{2}=32$$

Short Answer

Expert verified
r = 卤4

Step by step solution

01

- Isolate the Quadratic Term

Divide both sides of the equation by 2 to isolate the quadratic term: \[ \frac{2r^2}{2} = \frac{32}{2} \] Simplifies to: \[ r^2 = 16 \]
02

- Apply the Square Root Property

Take the square root of both sides of the equation: \[ \text{If } r^2 = 16, \text{ then } r = \text{卤} \sqrt{16} \] Simplifies to: \[ r = \text{卤} 4 \]

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Key Concepts

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

Isolating the Quadratic Term
To solve a quadratic equation using the Square Root Property, the first step is to isolate the quadratic term. Isolating the quadratic term means making sure that the term with the variable squared, here it's \(r^2\), is by itself on one side of the equation.
In our exercise, we have the equation: \[ 2r^2 = 32 \] To isolate the quadratic term \(r^2\), divide both sides of the equation by 2. This gives us: \[ \frac{2r^2}{2} = \frac{32}{2} \] This simplifies to: \[ r^2 = 16 \] Now, the quadratic term \(r^2\) is isolated and we can move to the next step.
Solving Quadratic Equations
Once the quadratic term is isolated, the next step is solving the quadratic equation. In the current context, we use the Square Root Property to solve it.
Quadratic equations are of the form \[ ax^2 + bx + c = 0 \] In our specific case, after isolating the quadratic term, we have: \[ r^2 = 16 \] This is a simplified form of a quadratic equation where there is no \(bx\) term or constant term \(c\) on one side.
At this stage, we take the square root of both sides of the equation to solve for \(r\).
Taking Square Roots
Taking the square root of both sides is the final step in using the Square Root Property. By doing so, you remove the exponent (which is 2 in this case) from the variable.
Starting from: \[ r^2 = 16 \] Take the square root of both sides: \[ r = \text{卤}\text{鈭殅16 \] The square root of 16 is 4, so we get: \[ r = \text{卤} 4 \] This means \(r\) can be either 4 or -4. Both values satisfy the original equation when squared.
Therefore, the solution to the equation is: \[ r = 4 \text{ or } r = -4 \]

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Most popular questions from this chapter

Solve. Round answers to the nearest tenth. A retailer who sells backpacks estimates that by selling them for \(x\) dollars each, he will be able to sell \(100-x\) backpacks a month. The quadratic function \(R(x)\) \(=-x^{2}+100 x\) is used to find the \(R,\) received when the selling price of a backpack is \(x .\) Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=x^{2}-6 x+8$$

Solve each inequality algebraically and write any solution in interval notation. $$2 x^{2}+5 x-12>0$$

Graph the function by using its properties. $$f(x)=-x^{2}+2 x-7$$

Graph each function using transformations. $$f(x)=(x-3)^{2}+4$$
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