Solving the Quadratic Equation: u² + 1 = 3u and Its Transformed Form

Understanding how to solve quadratic equations is a fundamental skill in algebra, essential for students and math enthusiasts alike. One common transformation in quadratic problems is rearranging expressions like u² + 1 = 3u into standard quadratic form u² - 3u + 1 = 0. This not only simplifies solving but also reveals important properties about the equation’s solutions. In this article, we’ll explore how to manipulate the equation, solve it using the quadratic formula, interpret the solutions, and apply these techniques in real-world scenarios.

Understanding the Context

Step 1: Rearranging the Equation

The original equation is:
u² + 1 = 3u

To solve for u, bring all terms to one side to form a standard quadratic equation:
u² - 3u + 1 = 0

This transformation is key because it allows direct application of quadratic solving methods.

Solved: 1. (u+3)/u-2 ≥slant 0 6. (u^2-5u+6)/u+4

Image Gallery

Solved Example Given u˙1=u2,u1(0)=1,u˙2=u1−2t,u2(0)=0, for | Chegg.com

Solved Let u1=[1-1-1],u2=[211], and u3=[001]. Note that u1 | Chegg.com

Solved Let u1=[11-2],u2=[5-12], and u3=[001]. Note that u1 | Chegg.com

Key Insights

Step 2: Identifying Coefficients

A standard quadratic equation is written as:
au² + bu + c = 0

Comparing this with our equation:

a = 1
b = -3
c = 1

🔗 Related Articles You Might Like:

📰 The secret sore she haunts every woman—searya’s silent curse no one talks about 📰 Shearya broke our hearts—what she says changes everything, but never share her name 📰 Why searia’s smile hides more than beauty—truth about the pain only she knows 📰 What Is A Word Processor 7805813 📰 Spider Man 1 Ps1 Cheats 7848486 📰 How To Factory Reset A Iphone 571582 📰 Finger Stock Price Explosionyou Wont Believe What Hit The Market Today 9288950 📰 Can You Recover Your Myapps Account Heres How To Log In Fast 4324346 📰 Twilight Saga Breaking Dawn Part 2 Cast 4412994 📰 5 Why Is Everyone Talking About Superior Iron Man The Iron Legends Just Got Unstoppable 1184992 📰 Los Tres Hermanos 6224644 📰 Faceit Finder Just Shook The Platform What It Found May Change Everything 7940173 📰 Find Old 401K 2861283 📰 Ptc Stock Price 4949511 📰 Fox Terrier Hair The Luxurious Coat That Steals Every Heart Claim It Now 9962122 📰 Windows Cloud Desktop Review Is It The Revolutionary Work Solution Youve Been Waiting For 3859934 📰 Num Lock Blocked You Heres The Shocking Way To Turn It Off Now 5407533 📰 You Wont Believe The Hidden Value Behind Elaborately Woven Jute Rugs 5237314

Final Thoughts

Step 3: Solving Using the Quadratic Formula

The quadratic formula solves for u when the equation is in standard form:
u = [ -b ± √(b² - 4ac) ] / (2a)

Substituting a = 1, b = -3, c = 1:
u = [ -(-3) ± √((-3)² - 4(1)(1)) ] / (2 × 1)
u = [ 3 ± √(9 - 4) ] / 2
u = [ 3 ± √5 ] / 2

Thus, the two solutions are:
u = (3 + √5)/2 and u = (3 − √5)/2

Step 4: Verifying Solutions

It’s always wise to verify solutions by substituting back into the original equation. Let’s check one:
Let u = (3 + √5)/2

Compute u²:
u = (3 + √5)/2 → u² = [(3 + √5)/2]² = (9 + 6√5 + 5)/4 = (14 + 6√5)/4 = (7 + 3√5)/2

Now, left side:
u² + 1 = (7 + 3√5)/2 + 1 = (7 + 3√5 + 2)/2 = (9 + 3√5)/2

Right side:
3u = 3 × (3 + √5)/2 = (9 + 3√5)/2