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Chapter 9: Problem 32

Solve equation by using the square root property. Simplify all radicals. \(3 x^{2}+8=80\)

Short Answer

Expert verified
\( x = \pm 2\sqrt{6} \)

Step by step solution

01

- Isolate the quadratic term

First, isolate the term involving the square by subtracting 8 from both sides of the equation: \[ 3x^2 + 8 - 8 = 80 - 8 \] which simplifies to \[ 3x^2 = 72 \]
02

- Divide by the coefficient of the squared term

Next, divide both sides of the equation by 3 to isolate the squared term: \[ \frac{3x^2}{3} = \frac{72}{3} \]This simplifies to \[ x^2 = 24 \]
03

- Apply the square root property

Now that the quadratic term is isolated, take the square root of both sides of the equation: \[ \sqrt{x^2} = \pm \sqrt{24} \]This becomes \[ x = \pm \sqrt{24} \]
04

- Simplify the radical

Lastly, simplify the radical \( \sqrt{24} \). Notice that \( \sqrt{24} \) can be simplified to \( 2\sqrt{6} \): \[ x = \pm 2\sqrt{6} \]

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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 method used to solve quadratic equations of the form \( ax^2 = c \). It involves taking the square root of both sides of the equation to solve for \( x \).

Here's how it works:
  • If \( x^2 = a \), then \( x = \pm\sqrt{a} \).
  • You get two solutions: one positive and one negative.

In our example, after isolating the quadratic term, the equation becomes \( x^2 = 24 \).
Taking the square root of both sides, we use the square root property: \( \sqrt{x^2} = \pm \sqrt{24} \).


This property is very useful because it directly gives us the solutions for quadratic equations without further factoring or completing the square.
Simplifying Radicals
Simplifying radicals helps make answers neater and easier to understand. A radical is in simplified form when it has no perfect square factors other than 1 inside the radical.

Let's break down the simplification of the radical \( \sqrt{24} \):
  • First, find the prime factorization of 24: \( 24 = 2 \times 2 \times 2 \times 3 \).
  • Group the factors into pairs: \( 2 \times 2 = 4 \).
  • Apply the square root to the grouped factors: \( \sqrt{4} = 2 \).

So, \( \sqrt{24} = 2 \sqrt{6} \).

This step is important as it ensures that the solution is fully simplified, making it more elegant and easier to interpret.
In our exercise, \( x = \pm 2 \sqrt{6} \) is the simplified form of \( x = \pm \sqrt{24} \).
Isolating the Quadratic Term
To solve quadratic equations using the square root property or other methods, start by isolating the quadratic term. This means getting the term with \( x^2 \) by itself on one side of the equation.

Here are the steps to isolate the quadratic term:
  • First, identify all terms in the equation. For example, in \( 3 x^2 + 8 = 80 \), the quadratic term is \( 3 x^2 \).
  • Move any constants to the other side of the equation by performing appropriate operations.
    In our example, subtract 8 from both sides: \( 3 x^2 + 8 - 8 = 80 - 8 \), which simplifies to \( 3 x^2 = 72 \).
  • Next, divide both sides by the coefficient of \( x^2 \).
    In this case: \( \frac{3 x^2}{3} = \frac{72}{3} \), simplifying to \( x^2 = 24 \).

Now, the quadratic term \( x^2 \) is isolated, and you can proceed to apply the square root property to solve for \( x \).

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Most popular questions from this chapter

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ x^{2}-4 x+5=0 $$

Solve each equation by completing the square. Give (a) exact solutions and (b) solutions rounded to the nearest thousandth. $$(x+1)(x+3)=2 $$

Evaluate the expression \(\sqrt{b^{2}-4 a c}\) for the given values of \(a, b,\) and \(c .\) Simplify the radicals. See Sections \(1.3,8.1,\) and 8.2 $$a=9, b=30, c=25$$

Use the quadratic formula to solve each equation. (a) Give solutions in exact form, and (b) use a calculator to give solutions correct to the nearest thousandth. $$ x^{2}=1+x $$

Solve equation by using the square root property. Simplify all radicals. \(5 x^{2}+4=8\)
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