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Chapter 0: Problem 95

Solve each quadratic equation by the method of your choice. $$3 x^{2}=60$$

Short Answer

Expert verified
Hence, solutions for the equation 3\(x^2 = 60\) are \(x = 2 \sqrt {5}\) and \(x = -2 \sqrt {5}\).

Step by step solution

01

Setting up the equation

Given the equation \(3x^2 = 60\). The task is to solve for \(x\).
02

Simplifying the equation

Divide both sides of the equation by 3 to make \(x^2\) isolated on one side. Which gives: \(x^2 = 60/3 = 20\).
03

Taking Square Root

Taking square root on both sides, we get \(x = \sqrt {20}\) and \(x = - \sqrt {20}\) as the quadratic equation has two roots.
04

Simplifying the Root

Further simplifying the root, \(x = \sqrt {20} = 2 \sqrt {5}\) and \(x = - \sqrt {20} = -2 \sqrt {5}\).

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Key Concepts

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

Square Root Method
The square root method is a simple way to solve quadratic equations when they are set up properly. This technique requires the quadratic equation to be in the form of \[ ax^2 = c \] where \( a \) and \( c \) are constants. The goal is to isolate \( x^2 \) on one side of the equation to make it straightforward to take the square root. To use the square root method:
  • First, ensure \( x^2 \) is isolated by dividing both sides of the equation by \( a \).
  • Once isolated, take the square root of both sides to solve for \( x \).
  • Remember that taking the square root yields two solutions: a positive and a negative root.
This method is efficient when the quadratic term is easily isolated and computable. If a quadratic fails to fit this easy form, consider completing the square or using the quadratic formula.
Quadratic Equations
Quadratic equations are polynomial equations of the second degree. They have the general form: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations are fundamental in algebra and appear frequently in various problem-solving scenarios.
To solve a quadratic equation, you have several methods at your disposal, including:
  • Factoring
  • Completing the square
  • Quadratic formula
  • Graphing
  • Square root method (when applicable)
Each method has its own use cases and can be selected based on the specific equation you are dealing with. It's all about finding the most efficient approach for the quadratic you are tackling.
Simplifying Square Roots
After using the square root method, you might get a result that needs further simplification. This involves expressing the root in its simplest "radical" form. For example, consider \( \sqrt{20} \). To simplify a square root:
  • Identify perfect square factors of the number under the root. For 20, it's 4, because \( 4 \times 5 = 20 \).
  • Break down the expression: \( \sqrt{20} = \sqrt{4 \times 5} \).
  • Use the property \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) to simplify: \( \sqrt{20} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \).
This technique reduces the number under the radical to its simplest form, making it easier to interpret and use in equations. Simplification is important, as it helps in comparing roots, solving further algebraic expressions, and ensuring precision in mathematics.

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. If \(y=|x+1|,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -4 and ending with 2

What is a rational expression?

Find \(b\) such that \(\frac{7 x+4}{b}+13=x\) will have a solution set given by \(\\{-6\\}\).

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$ \text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. } $$ How old is the son?b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

What is a quadratic equation?
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