Problem 12 Solve each equation. $$ (q-2... [FREE SOLUTION]

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McGraw Hill

$Math Studyset 91影视 Explanations$ Math

2004 Edition

Chapter 7: Problem 12

Solve each equation. $$ (q-2)^{2}+8=44 $$

Short Answer

Expert verified

The solutions are $ q = 8 $ and $ q = -4 $.

Step by step solution

Isolate the squared term

First, subtract 8 from both sides of the equation to isolate the squared term. \[(q-2)^{2} + 8 - 8 = 44 - 8\] This simplifies to \[(q-2)^{2} = 36\]

Take the square root of both sides

Next, take the square root of both sides to solve for $q-2$. \[\begin{aligned} \text{If} \, (q-2)^2 = 36, \, \text{then}\ q-2 &= \ \text{positive or negative square root of 36} \ &= \ q-2 &= \ \text{6 or -6} \end{aligned}\] Thus, \[ q-2 = 6 \quad \text{or} \quad q-2 = -6 \]

Solve for q

Finally, solve for $q$ from each equation. \[ q-2 = 6 \Rightarrow q = 6 + 2 = 8\] \[ q-2 = -6 \Rightarrow q = -6 + 2 = -4\] So, the solutions are \[ q = 8 \quad \text{and} \quad q = -4 \]

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Key Concepts

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

Isolate the Squared Term

When solving quadratic equations, the first step often involves isolating the squared term. This means getting the term that contains the square, $ (q-2)^2 $, on one side of the equation by itself. In the given exercise, we start with: $ (q-2)^{2}+8=44 $. The goal is to move the constant, +8, to the other side.

To do this, we subtract 8 from both sides:

\[ (q-2)^{2}+8-8=44-8 \]

Simplifying that, we isolate the squared term: \[ (q-2)^{2}=36 \]

This preparation is crucial for the next steps.

Taking Square Roots

After isolating the squared term, the next step is to remove the square by taking the square root of both sides. This is essential for solving $ q $ because it simplifies the equation.

For our equation \[ (q-2)^{2}=36 \], we take the square root of both sides:

\[ \text{If} \, (q-2)^2 = 36, \, \text{then} \, q-2 = \, \text{positive or negative square root of 36} \]

The square root of 36 has two values because both +6 and -6, when squared, result in 36.

Therefore, we have:

\ q-2 = 6 \
\ q-2 = -6 \

Now we have two possible equations to solve for \ q \.

Solving for Variables

The final step in solving a quadratic equation is to solve for the variable $ q $. With the two equations:

\ q-2=6 \
\ q-2=-6 \

We isolate \ q \ by adding 2 to both sides for each equation:

For \ q-2=6 \:
\ q = 6 + 2 \
\ q = 8 \

For \ q-2=-6 \:
\ q = -6 + 2 \
\ q = -4 \

Thus, the two solutions are: \ q = 8 \ and \ q = -4 \.

These steps demonstrate how to handle the different parts of the quadratic equation effectively.

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91影视

Solve each equation. $$ (q-2)^{2}+8=44 $$

Short Answer

Step by step solution

Isolate the squared term

Take the square root of both sides

Solve for q

Key Concepts

Isolate the Squared Term

Taking Square Roots

Solving for Variables

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