y(y^2 - 5y + 6) = 0 \Rightarrow y(y - 2)(y - 3) = 0 - Get link 4share
Understanding the Equation: Solving y(y² – 5y + 6) = 0 and Why It Equals y(y – 2)(y – 3) = 0
Understanding the Equation: Solving y(y² – 5y + 6) = 0 and Why It Equals y(y – 2)(y – 3) = 0
When solving quadratic and polynomial equations, factoring plays a crucial role in simplifying expressions and finding solutions efficiently. One classic example is the equation:
y(y² – 5y + 6) = 0
But why is this equation rewritten as y(y – 2)(y – 3) = 0? And how does factoring help in finding the roots? In this SEO-optimized article, we’ll explore step-by-step how this transformation works and how factoring enables us to solve the equation quickly using the zero-product property.
Understanding the Context
What Does y(y² – 5y + 6) = 0 Mean?
The original equation
y(y² – 5y + 6) = 0
is a product of two factors:
- Main factor:
y - Quadratic factor:
y² – 5y + 6
Since the product equals zero, the zero-product property tells us that at least one of the factors must be zero:
❌ y = 0
✅ y² – 5y + 6 = 0
Key Insights
This leads us to solving the quadratic, but factoring the quadratic trinomial makes the solution elegant and fast.
Factoring y² – 5y + 6 into (y – 2)(y – 3)
To rewrite y² – 5y + 6, we look for two numbers that:
- Multiply to +6 (the constant term)
- Add to –5 (the coefficient of the middle term)
The numbers –2 and –3 satisfy these conditions:
– (–2) × –3 = –6? Wait — correction:
Actually, —2 × –3 = +6 ✅ and
–2 + –3 = –5 ✅
🔗 Related Articles You Might Like:
📰 Unlock the Hidden Stories Behind Every Jungle Animal’s Secret Powers 📰 The Ultimate Guide to Jungle Characters You Thought You Knew—Now Radical Reveals 📰 The Ultimate Bow Wallpaper You've Never Seen Before 📰 This Rare Taurus Scorpio Match Will Change Your Life Forever Mind Blowing Compatibility 📰 This Rare Tea Cup Hidden In Action Changes How You Drink Tea Forever 📰 This Rare Trick In Super Mario 64 Will Make You Replay The Game From Day 1 📰 This Retro Classic Changed Gaming Forever See What Made Super Mario Bros 3 Unforgettable 📰 This Revolutionary Ingredient In Sulfler Is Changing How We Look At Skincare Forever 📰 This Revolutionary Stufful Will Blow Your Mindwatch Now To Discover 📰 This Rise To Fame Was Unstoppable Find Out Why Everyones Talking About Tenzin 📰 This Risky Switch Just Made Me Rich Overnight Discover The Switch 2 Stock Survival Guide 📰 This Safer Alternative To Standard Ceiling Tiles Is Changing Home Renovations Forever 📰 This Savage Tattoo On Vagina Cul App Hindu Shocking Styles Pain Or Pleasure 📰 This Scandalous Truth About Suzanne Perry Will Change How You See Her Legacy Forever 📰 This Scary New Subscriber Identity Module Card Could Ruin Your Online Accounts Now 📰 This Season 13 Of Supernatural Will Shock Youyou Wont Believe What Happens Next 📰 This Season 5 Of Temptation Island The Ultimate Betrayal Youll Rip Through 📰 This Season Of Stranger Things 3 Is Blowing Upheres What You MissedFinal Thoughts
Thus,
y² – 5y + 6 = (y – 2)(y – 3)
Substitute back into the original expression:
y(y² – 5y + 6) = y(y – 2)(y – 3) = 0
Now the equation is fully factored and clearly shown.
Why Is Factoring Important in Solving Equations?
Factoring transforms a potentially complex expression into a product of simpler binomials. Once factored, applying the zero-product property becomes straightforward — each factor is set to zero, unlocking the solutions.
For y(y – 2)(y – 3) = 0,
set each factor equal to zero:
- y = 0
- y – 2 = 0 → y = 2
- y – 3 = 0 → y = 3
Thus, the solutions are y = 0, y = 2, and y = 3
