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 - 500apps
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:
📰 Truly Breathtaking Flowers: The Eyes-Wide-Opening Bloom Show You Need to See! 📰 Eye Black Secrets: How NBA Stars Use This Dark Magic to Blow Their Opponents Away! 📰 Shade That Glare: The Ultimate Eye Black Tricks Every Athlete Needs! 📰 Shocking Beauty Inside The Long Haired Alsatian Dog That Steals Every Heart 📰 Shocking Benefits Of Using A Luffa Sponge You Need To Try Today 📰 Shocking Best Long Sleeve Grad Dresses Every Fashion Lover Needs To Try Now 📰 Shocking Breakdown Low Taper Fade Curly Hair Looks Easier Than You Thinktry It Now 📰 Shocking Breakdown Of The Most Stunning Manga Panel Every Fan Needs To Check 📰 Shocking Breakdown When Malign Sparks Urgent Action Learn The Warning Signs Today 📰 Shocking Client Reactions At M Bird Salonwatch How One Magic Makeover Cost 500 📰 Shocking Confession From Matt Maher Lord I Need Youdont Miss This Mind Blowing Moment 📰 Shocking Connection Between Majoras Mask And The Moon You Need To Watch This 📰 Shocking Deal Logitech Loudspeakers Are Reducing Noise Boosting Clarity 📰 Shocking Deal Purebred Maltipoo For Salespot Yours Before Its Gone 📰 Shocking Design Timeless Power The Ultimate Manga Basilisk Guide Revealed 📰 Shocking Detail That Made This Lotus Bridal Dress Instantly Stunningyoull Want It 📰 Shocking Details Revealed The Mario Luigi Brothership That Apps Just Cant Ignore 📰 Shocking Discoveries About Martin Mulls Movies And Tv Shows You Never Saw ComingFinal 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.
