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Chapter 9: Problem 31

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

Short Answer

Expert verified
t = 卤 3鈭�3

Step by step solution

01

- Isolate the quadratic term

First, you need to isolate the term that contains the quadratic expression. Start by subtracting 7 from both sides of the equation:t2 + 2t2 + 7 - 7 = 61 - 7This simplifies to:t2 + 2t2 = 54
02

- Simplify the equation

Next, divide both sides by 2 to isolate the square term:2t2 + 2t2/2 = 54/2t2 + 2t2 = 27
03

- Apply the square root property

The square root property states that if t2 = t, then t = t2 卤 t2. Apply this property:t2 = 27Therefore,t = at27 卤 at27
04

- Simplify the radical

Simplify the square root expression. We have:tr=3t3So, the final solution is:[t = t3, {卤 3t3, 卤 3t3}]

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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 valuable tool for solving quadratic equations. It states that if you have an equation of the form \[x^2 = k\], you can solve for x by taking the square root of both sides: \[x = \pm \sqrt{k}\]. This property is very useful when you have isolated the quadratic term and want to quickly find the solutions. Remember, since both positive and negative values of x will satisfy the equation, always include the \pm sign. By understanding this property, you can solve many quadratic equations effectively.
isolating the quadratic term
The first step in solving a quadratic equation using the square root property is isolating the quadratic term. In our exercise, we start with the equation:\[2 t^{2} + 7 = 61\].
  • Subtract 7 from both sides to eliminate the constant term on the left side:
    • \[2 t^{2} + 7 - 7 = 61 - 7\].
  • This simplifies to:
    • \[2 t^{2} = 54\].
Now, we have successfully isolated the quadratic term by getting rid of the constant on the left side. This step sets the stage for applying the square root property.
simplifying radicals
Simplifying radicals is key when solving quadratic equations. In our example, once we have the equation \[t^2 = 27\], we take the square root of both sides:\[t = \pm \sqrt{27}\].We can further simplify \sqrt{27} by noting that 27 is equal to \[3^2 \times 3\]. So, \sqrt{27} = \sqrt{3^2 \times 3} = 3 \sqrt{3}\. Hence, the term \sqrt{27} simplifies to \[3 \sqrt{3}\].Therefore, our solutions are:\[t = \pm 3 \sqrt{3}\].
By simplifying radicals properly, we ensure that our solutions are in their simplest form.
step-by-step solution
Let's go through the solution step-by-step to make everything clear. Step 1: Isolate the quadratic term
Subtract 7 from both sides: \[2 t^{2} + 7 - 7 = 61 - 7\].
This simplifies to: \[2 t^{2} = 54\].
Step 2: Divide by coefficient of \[t^2\]
Divide both sides by 2 to get \[t^2\]: \[t^{2} = 27\].
Step 3: Apply the square root property
Take the square root of both sides: \[t = \pm \sqrt{27}\].
Step 4: Simplify the radical
Simplify \sqrt{27} to get \[3 \sqrt{3}\] resulting in: \[t = \pm 3 \sqrt{3}\].
Following these steps ensures that you find the correct solutions
[\(t = 3 \sqrt{3}\) and \(t = -3 \sqrt{3}\)].

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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.

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