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.

$u^2 + 1 = 3u \Rightarrow u^2 - 3u + 1 = 0$

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:

📰 "This Pikachu Detective Just Solved the Mysterious Case Involving Gym Leaders! Are You Ready?! 📰 "Pikachu is a Detective Now—Can You Solve the Ultimate Pokémon Mystery?! 📰 "Shocking Twists in Pokemon Detective Pikachu’s Latest Case You Can’t Believe! 📰 The Crown Of Glorythis Club Champion Wasnt Supposed To Return 📰 The Crown Of Midnight Burnswhat Truth Lies Trapped Within Its Gloom 📰 The Crown Of Midnight Hides A Curse No One Was Meant To See 📰 The Cruise Mapper That Will Turn Your Vacation Into A Hidden Adventure 📰 The Crushing Truth Behind Barrys Bootcamp That Most Ignore 📰 The Crystal Ball Shatters Your Future See Whats Coming 📰 The Cult Of Chucky Strikes Backthe Terror Has Returned 📰 The Cult That Conjured Four What Hoped To Gainnow Theyve Lost Everything 📰 The Cumberland Times Isnt Reportingits Hiding What Matters 📰 The Curse Of Chucky Engulfs The Innocentwatch Horror Unravel 📰 The Cute Stickers That Are Taking Over Your Life 📰 The Cutest Christmas Wallpaper That Will Brighten Your Holiday Season Forever 📰 The Cutest Emojis Thatll Brighten Any Message Instantly 📰 The Cutest Skies That Will Make You Feel Like Youre In A Fairy Tale 📰 The Cutest Wallpapers That Steal Attention Instantlyyour New Phone Backdrop Is Here

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