\fracn2 (4n + 10) = 150 \implies n(4n + 10) = 300 \implies 4n^2 + 10n - 300 = 0 - Decision Point
Solving the Quadratic Equation: \frac{n}{2}(4n + 10) = 150 – Step-by-Step Guide
Solving the Quadratic Equation: \frac{n}{2}(4n + 10) = 150 – Step-by-Step Guide
If you've ever encountered an equation like \(\frac{n}{2}(4n + 10) = 150\), you know how powerful algebra can be when solving for unknown variables. This article breaks down how to solve this quadratic equation step by step, showing how to transform, simplify, and apply the quadratic formula to find accurate values of \(n\).
Understanding the Context
Understanding the Equation
We begin with:
\[
\frac{n}{2}(4n + 10) = 150
\]
This equation suggests a proportional relationship multiplied by a linear expression, then set equal to a constant. Solving this will help us uncover the value(s) of \(n\) that satisfy the equation.
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Key Insights
Step 1: Eliminate the fraction
To simplify, multiply both sides of the equation by 2:
\[
2 \cdot \frac{n}{2}(4n + 10) = 2 \cdot 150
\]
\[
n(4n + 10) = 300
\]
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Now we expand the left-hand side.
Step 2: Expand the quadratic expression
Distribute \(n\) across the parentheses:
\[
n \cdot 4n + n \cdot 10 = 4n^2 + 10n
\]
So the equation becomes:
\[
4n^2 + 10n = 300
\]
Step 3: Bring all terms to one side
To form a standard quadratic equation, subtract 300 from both sides:
