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 - NBX Soluciones
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:
📰 Satellite Text Verizon 📰 My Verizon Wireless Business Login 📰 Verizon Login Cloud 📰 Fighting Type Are Weak Against 7321789 📰 Discover The Codes That Make Anime Rangers X Unstoppable Dont Miss These 1312628 📰 Cannot Stop Watching The Best Sean Williams Scott Movies That Set Hollywood On Fire 695457 📰 Are Romanian White 8905554 📰 Plumbenefits You Never Knew Existed That Could Rewire Your Plumbing Forever 6240967 📰 Best Travel Credit Card Mastercard 6059185 📰 Judge Dismisses Military Trespass Charges Against Migrants In New Mexico 1495826 📰 Speak Any Language Instantlyfree Iphone App Thats A Game Changer 4680803 📰 Free Shirts For Roblox 2331024 📰 Sandra Dee Cause Of Death 8731399 📰 5 Stop Confusing Retirement Goldira Vs 401K Explained You Wont Guess The Key Difference 2893960 📰 Kiana Madeira 3268378 📰 Surface Pro Laptops The Ultimate Upgrade Nobody Was Expectingsee Why They Dominating Tech Claims 9362486 📰 Jasmine Crockett Removed 9102566 📰 Visor De Imagenes Windows 10 6964924Final 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.
