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Chapter 6: Problem 10

Simplify and reduce each expression. $$ \frac{-8 \pm \sqrt{72}}{4} $$

Short Answer

Expert verified
The simplified expressions are \(-2 + \frac{3\sqrt{2}}{2}\) and \(-2 - \frac{3\sqrt{2}}{2}\).

Step by step solution

01

Calculate the Square Root

Start by simplifying the square root in the expression. We have \( \sqrt{72} \), which can be simplified to \( \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \).
02

Substitute Back into the Expression

Now, substitute \( 6\sqrt{2} \) back into the expression to get: \[\frac{-8 \pm 6\sqrt{2}}{4}\]
03

Simplify the Fraction

Simplify the fraction by dividing each term in the numerator by the denominator. The expression becomes: \[\frac{-8}{4} \pm \frac{6\sqrt{2}}{4} = -2 \pm \frac{3\sqrt{2}}{2}\]
04

Final Simplified Expression

Combine the terms to get the final simplified expressions as: \[-2 + \frac{3\sqrt{2}}{2} \quad \text{and} \quad -2 - \frac{3\sqrt{2}}{2}\]

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Key Concepts

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

Quadratic Formula
The quadratic formula is a tool used to find solutions to quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). It is expressed as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This powerful formula allows us to directly calculate the roots of any quadratic equation, without factoring or graphing.
Let's break it down:
  • 'a', 'b', and 'c' are coefficients in the quadratic equation.
  • The term \(b^2 - 4ac\) is called the discriminant, and it determines the nature of the roots.
  • The plus-minus symbol \( \pm \) indicates there are generally two solutions.
Quadratic equations describe parabolas when graphed, and the roots are the x-intercepts. Using the quadratic formula is often a straightforward pathway to finding these intercepts, especially when other methods are cumbersome.
Square Roots
Understanding square roots involves finding a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\).
Simplifying square roots can make them easier to work with in math problems. For instance, \(\sqrt{72}\) can be simplified by factoring inside its radicand:
  • Find perfect squares within the number: \(72\) can be broken down into \(36 \times 2\).
  • The square root of a product can be split: \(\sqrt{36} \times \sqrt{2} = 6\sqrt{2}\).
Doing this helps in simplifying expressions and solving equations. Remember, not all numbers simplify nicely (i.e., \(\sqrt{2}\) remains \(\sqrt{2}\)), but knowing how to factor smartly can simplify your calculations greatly.
Fraction Simplification
Fraction simplification involves reducing the fraction to its simplest form. By dividing both the numerator and the denominator by their greatest common factor (GCF), you can simplify the fraction.
Let's consider the expression \(\frac{-8 \pm 6\sqrt{2}}{4}\):
  • Both terms in the numerator can individually be divided by 4, the denominator.
  • This results in \(-2\) for \(\frac{-8}{4}\) and \(\frac{3\sqrt{2}}{2}\) for \(\frac{6\sqrt{2}}{4}\).
The simplification process helps in keeping expressions neat and manageable. It's an important skill in algebra that ensures your answers are as tidy and precise as possible.

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Most popular questions from this chapter

Set up an equation and solve each problem. A rectangular plot of ground measuring 12 meters by 20 meters is surrounded by a sidewalk of a uniform width (see Figure 6.9). The area of the sidewalk is 68 square meters. Find the width of the walk.

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Solve each inequality. $$ x^{2}-14 x+49 \geq 0 $$
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