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 - inBeat
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:
📰 You Wont Believe What This Special Tax Could Mean for Your Wallet! 📰 Special Tax Alert: Government Just Unleashed a Billionaires Nightmare! 📰 Special Tax? Heres How It Could Slap You with the Biggestimpact Yet! 📰 Caitlin Clark Shower Sneak Peek That Broke Social Mediawhat Did She Do 3062330 📰 Breakdown Inside Perfect For Your Next Ink Hurry And Inspire 1918861 📰 Bank It Jobs 7428372 📰 See The Most Stunning Floral Day Of The Dead Displays That Will Change Your Perspective 7165476 📰 Wcgn Stock Price Is Energy But Investors Are Going In Bare Percentages 2158921 📰 Shibarium News 1464516 📰 Bernice Julien Shocked Everyoneher Secret Behind Her Astounding Rise To Fame 5757421 📰 Production Year 2013Present 6830288 📰 Giovanni Pelletier 4282775 📰 Footdrop Alert The 825 Area Code Is Taking Over Now Dont Miss These Big Changes 2550594 📰 Step By Step Guide Where To Report Hipaa Violations And Protect Your Privacy Fast 5976189 📰 American Express Gold Card 7810494 📰 The Trade Desk News Today Shocking Increase In Digital Ad Spending Stuns The Industry 4740560 📰 The Untold Story Of Surface 1943 How One Object Changed History Forever 6320363 📰 Standard Deduction 2023 1124682Final 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.
