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:

📰 You Won’t Believe How Happy This Friday Looks – Breathtaking Images to Brighten Your Mood! 📰 Happy Friday 2024: Watch These Joyful Photos Go Viral Overnight! 📰 Joy Starts Now: The Most Happy Friday Moments Captured in Stunning Images! 📰 Nacho Libres Ultimate Costume Demons Every Fans Wild Fear 📰 Nachro Vidals Obsession Unveiled The Epic Scent Of His 25 Liter Perfume Bottle You Wont Believe It Came In 📰 Nad Injections The Miracle Secret Doctors Refuse To Tell You 📰 Nad Is The Hidden Key To Inner Energydid You Miss It 📰 Nad Ni Poziom Morza Tajemnica Ukrytamsg Szivplets Ogromnej Ciefy 📰 Nad Si Poziomem Morza Zlayer Ziemi Kracza Oku Ni Orzechw 📰 Nag Champa Exposed Something That Changed How We See Her Forever 📰 Nag Champas Secret That Will Make You Crave Her Again Forever 📰 Naga Jolokias Ghost Haunts Mysterious Corners Of Darkness 📰 Nail Colour So Stunning It Will Steal Your Heart This Autumn You Wont Look Away 📰 Nails So Perfect Theyll Rival Professional Artisanswatch How 📰 Naked And Blurry The Unsettling Truth Behind The Photoshoot Cover Up 📰 Naked And Unaware Moms Private Moment Becomes Internet Fever 📰 Naked Capitalism Exposes What Supply Chains Wont Let You See 📰 Naked Capitalism When Profit Has No Dexterity Or Conscience

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