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:

📰 Download Windows 10 Media Creation Tool 📰 Download Windows 10 on Flash Drive 📰 Download Windows 10 on Usb Drive 📰 Veizon Internet 📰 Stop Guessing Heres The Secret Way To Sign Your Word Documents 6670366 📰 Discover The Best Clear Phone Case That Drops Zero Screen Shadows 474147 📰 Bodoni 6433302 📰 Brackets App 📰 St Chroma Lyrics 7247637 📰 3 From Vampire To Hellbringer Alucard Hellsings Epic Journey Unveiled 9331920 📰 Better Kid Care Login 📰 Fha Home Loan Mortgage 3759886 📰 Citicards Online 📰 Roblox Studios Create 📰 Mad Cartoon Character 📰 Kid3 Tag Editor 📰 Breaking Gainesville Times Reveals Hidden Secrets Of Local Government Shockingly 1746957 📰 Fire Retirement Unlocked The Shocking Ways To Maximize Your Savings Now 3676076

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