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 - Tacotoon
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:
📰 "From Justice to Madness: These DC Women Are Redefining Heroism! 📰 "Secrets Behind DC’s Best Female Heroes – Hard to Ignore Power & Mystery! 📰 🌟 DCCinema Unleashed: The Epic Truth About DC Animated Films You Didn’t Know! 📰 Trapped In The Leatherface Legacy Heres Everything Katara Avatars Secret Changes 📰 Travel To Leavenworth Washington This Christmas For The Most Enchanting Winter Wonderland 📰 Travis Scott Kylie Jenners Spontaneous Night This Secret Moment Is Going Viral 📰 Trend Alert The Ultimate Black Ladies Work Pants Everyones Raving Over Shop Now 📰 Truth About Lamashtu The Terrifying Truth Behind This Ancient Spirit 📰 Try The Next Largest Divisor 1012 20242 Can We Have A 1012 B 1012 Then A B 2024 Both Even And Gcd10121012 1012 📰 Try These Deadly Lateral Lunges Theyre Changing How Athletes Train 📰 Try These Undeniable Kielbasa Recipes That Will Turn Every Meal Into A Feast 📰 Turbulent Shadows How The League Of Assassins Is Taking Over The Game 📰 Turn Back The Clock Channel Iconic 70S Ladies Costumes Steal The Spotlight 📰 Turn Convulsion At Kashimo Jjc The Untold Truth Behind Monsters Magic 📰 Turn Ones Into Treasures Why This Keepsake Box Is A Time Capsule Of Forever 📰 Turn Tears Into Triumph Perfect Kindergarten Math Worksheets For Instant Success 📰 Turn Your Legs Into Art The Most Viral Leg Tattoo Trends For Women 📰 Turn Your Locks Into A Sensation With Kinky Hair PerfectionFinal 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.
