Question
Solve the equation. Express numbers in exact simplest form. \[ 8\left(q^{2}-2 q\right)=7 q^{2}-64 \] The solution set is \( \{\square\} \).
Ask by Campbell Hobbs. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( q = 8 \).
Solution
We start with the equation
\[
8\left(q^{2} - 2q\right) = 7q^{2} - 64.
\]
**Step 1. Expand and simplify the equation**
First, distribute \(8\) on the left-hand side:
\[
8q^{2} - 16q = 7q^{2} - 64.
\]
Subtract \(7q^{2}\) and add \(64\) to both sides to set the equation to zero:
\[
8q^{2} - 16q - 7q^{2} + 64 = 0.
\]
Combine like terms:
\[
q^{2} - 16q + 64 = 0.
\]
**Step 2. Factor the quadratic equation**
We observe that the quadratic expression is a perfect square:
\[
q^{2} - 16q + 64 = \left(q - 8\right)^{2}.
\]
Thus, the equation becomes:
\[
\left(q - 8\right)^{2} = 0.
\]
**Step 3. Solve for \(q\)**
Since \(\left(q - 8\right)^{2} = 0\), we have:
\[
q - 8 = 0.
\]
Solving for \(q\) gives:
\[
q = 8.
\]
The solution set is \( \{8\} \).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
First, let's start by expanding and rearranging the given equation: \[ 8(q^2 - 2q) = 7q^2 - 64 \] Distributing on the left side gives: \[ 8q^2 - 16q = 7q^2 - 64 \] Now, let's bring all terms to one side of the equation to set it to zero: \[ 8q^2 - 7q^2 - 16q + 64 = 0 \] This simplifies to: \[ q^2 - 16q + 64 = 0 \] Next, we can factor this quadratic equation: \[ (q - 8)(q - 8) = 0 \] This means: \[ (q - 8)^2 = 0 \] So, we have a double root: \[ q = 8 \] Thus, the solution set is: \[ \{8\} \] In exact simplest form, the solution is \( \{8\} \).
