n = \frac{-5 \pm \sqrt5^2 - 4(2)(-150)}2(2) = \frac{-5 \pm \sqrt25 + 1200}4 = \frac{-5 \pm \sqrt1225}4 = \frac-5 \pm 354 - Coaching Toolbox
Solving Quadratic Equations: A Step-by-Step Guide Using the Quadratic Formula
Solving Quadratic Equations: A Step-by-Step Guide Using the Quadratic Formula
Mastering quadratic equations is essential in algebra, and one of the most powerful tools for solving them is the quadratic formula:
\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Understanding the Context
In this article, we walk through a practical example using the equation:
\[
n = \frac{-5 \pm \sqrt{5^2 - 4(2)(-150)}}{2(2)}
\]
This equation models real-world problems involving area, projectile motion, or optimization—common in science, engineering, and economics. Let’s break down the step-by-step solution and explain key concepts to strengthen your understanding.
Image Gallery
Key Insights
Step 1: Identify Coefficients
The general form of a quadratic equation is:
\[
an^2 + bn + c = 0
\]
From our equation:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -150 \)
Plugging these into the quadratic formula gives:
\[
n = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-150)}}{2(2)}
\]
Step 2: Simplify Inside the Square Root
Simplify the discriminant \( b^2 - 4ac \):
\[
(-5)^2 = 25
\]
\[
4 \cdot 2 \cdot (-150) = -1200
\]
\[
b^2 - 4ac = 25 - (-1200) = 25 + 1200 = 1225
\]
So far, the equation reads:
\[
n = \frac{5 \pm \sqrt{1225}}{4}
\]
🔗 Related Articles You Might Like:
📰 why do people smoke 📰 left eye pain 📰 implantation bleeding pics 📰 Typescript News 5473028 📰 5Ly Temple Run Fan W Front Infinite Life Hack Thats Taking The Gaming World Fonic 7865127 📰 Akiyama Shock How This Icon Stunned Fans Worldwide Overnight You Wont Believe The Truth 3824044 📰 Sgv Newspaper Uncovers The Hidden Truthyoull Drop Everything To Read This 579727 📰 1800 Orleans St Baltimore Decades Of Mystery You Cant Ignore 1283613 📰 Why Walmart Failed In Germany 1690601 📰 Copilot Vs Chatgpt 6433336 📰 Will It Run On My Pc 4616811 📰 Citizenm Miami Brickell 3237836 📰 Rockstargames Socialclub 8185424 📰 Where To Watch Demon Slayer Infinity Castle 1130336 📰 Bills Practice Squad 1884172 📰 1St South Financial 9008828 📰 Watch The Linkin Park Logo Transform Every Song Into A Masterpiecewhat They Wont Show You 7594429 📰 Short Black Female Hair 3298562Final Thoughts
Step 3: Compute the Square Root
We now simplify \( \sqrt{1225} \). Since \( 35^2 = 1225 \),
\[
\sqrt{1225} = 35
\]
Now the expression becomes:
\[
n = \frac{-5 \pm 35}{4}
\]
(Note: Because \( -b = -(-5) = 5 \), the numerator is \( 5 \pm 35 \).)
Step 4: Solve for the Two Roots
Using the ± property, calculate both solutions:
1. \( n_1 = \frac{-5 + 35}{4} = \frac{30}{4} = \frac{15}{2} = 7.5 \)
2. \( n_2 = \frac{-5 - 35}{4} = \frac{-40}{4} = -10 \)
Why This Method Matters
The quadratic formula provides exact solutions—even when the discriminant yields a perfect square like 1225. This eliminates errors common with approximation methods and allows precise modeling of physical or financial systems.
Applications include maximizing profit, determining roots of motion paths, or designing optimal structures across STEM fields.
