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Chapter 0: Problem 95

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

Short Answer

Expert verified
The solutions of the given quadratic equation are \( x \approx 4.47 \) and \( x \approx -4.47 \).

Step by step solution

01

Bring the equation to standard form

The initial equation is \( 3x^{2} = 60 \). The first step is to isolate \( x^{2} \) by dividing both sides of the equation by \( 3 \). This gives us the equation \( x^{2} = \frac{60}{3} \).
02

Simplify the equation

By simplifying the right-hand side, we get \( x^{2} = 20 \). This means that \( x \) is equal to the square root of \( 20 \).
03

Find the square root

Because a square root has two possible values, \( x \) will have two possible values as well. So, \( x = \sqrt{20} \) and \( x = -\sqrt{20} \).
04

Simplify the square root

Taking the square root of \( 20 \), we have \( x = \approx 4.47 \) and \( x = \approx -4.47 \).

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Key Concepts

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

Standard Form
The standard form of a quadratic equation is essential for solving and understanding these types of problems. Typically, the standard form is expressed as \( ax^2 + bx + c = 0 \). It is like a blueprint for quadratic equations, allowing us to tackle them in a structured way. In the context of our example, we start with the equation \( 3x^2 = 60 \). To get it into standard form, we need to bring all terms over to one side of the equation.
  • This involves moving anything not involving \( x^2 \) to the opposite side.
  • However, for our current problem, since it's already about squares and equals without a linear term, our target is to prepare it for further methods.
  • Dividing by 3 simplifies it to \( x^2 = 20 \), which now fits the standard quadratic setup without distraction from linear components.
By grasping the standard form, we have a clear path to solve, simplifying the task ahead.
Square Root Method
The square root method provides a direct and straightforward way to solve quadratic equations once they're in the right form. This method is particularly helpful when the equation only contains an \( x^2 \) term and a constant on the other side.
  • If the equation is in the form \( x^2 = k \), we can solve for \( x \) simply by taking the square root of both sides.
  • It's crucial to remember that a square root can yield two solutions: a positive and a negative result.
  • In our example, \( x^2 = 20 \), taking the square root of 20 gives us two potential solutions.
Thus, you have \( x = \sqrt{20} \) and \( x = -\sqrt{20} \). This method is effective for equations like these, offering a quick resolution to quadratic problems in a familiar square setup.
Simplifying Equations
Simplifying equations is a fundamental part of solving quadratic equations, allowing us to work with more manageable numbers and expressions. Simplifying often means breaking down complex numbers into simpler components or factoring expressions to their simplest forms.
  • Initially, you may encounter an equation like \( 3x^2 = 60 \).
  • The first step towards simplicity is isolating \( x^2 \), done by dividing by 3 to get \( x^2 = 20 \).
  • Further simplification involves breaking down the square root, here \( \sqrt{20} \) simplifies to \( 2\sqrt{5} \).
Instead of a decimal approximation, breaking down \( \sqrt{20} \) offers clarity in precise terms, revealing the exact nature of the roots. By simplifying, we not only make calculations easier but also gain deeper insights into the properties of the equation itself.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although \(20 x^{3}\) appears in both \(20 x^{3}+8 x^{2}\) and \(20 x^{3}+10 x\) I'll need to factor \(20 x^{3}\) in different ways to obtain each polynomial's factorization.

Describe ways in which solving a linear inequality is different than solving a linear equation.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?

Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial.
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