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:

📰 Do You Know the Shocking Power of Burnin Rubber? Try It Now! 📰 Burnin Rubber Unleashed: The Game-Changer Developers Arent Wanting You To See! 📰 Burnin Rubber 5 XS: The Shockingly Durable Burn 📰 5 The Forgotten Samurai Game Legend Few Knowyoshimitsus Rise Is Sensational 3319135 📰 Skoliosexual 2094380 📰 This Neural Overload Game Stealed My Brainyou Wont Stop Thinking After Playing 5692489 📰 Air Fryer Corn Dogs This One Simple Trick Makes Them Unstoppable 3052866 📰 Drips By Pepsi New Soda Flavors 2638161 📰 Barrel Of Oil Price Now 1091772 📰 Exclusion Oig 4355797 📰 The Hidden Xbox Game Pass Games Everyones Too Busy To Discover Click To See Them All 6504793 📰 Inizio Pizza 8499485 📰 The 1 Free Online Game Thats Shocking How Many Players Are Hooked 7754544 📰 No Download No Problem Crazy Free Car Games Ready To Play Right Now 4101305 📰 Radius Nuclear Blast Unleashed How Far Can Destruction Spread 2077018 📰 3 Moviesjoyplus Revolves Around Epic Escapes Watch Your Favorite Flicks Now 6481263 📰 These 7 Hottest Anime Characters Cut Straight Through The Competition 7286012 📰 Finally A Smarter Way To Translate Korean No Fluff Just Results 8235892

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