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Understanding the Quadratic Equation 3t² + 11t + 10: A Comprehensive Guide
Understanding the Quadratic Equation 3t² + 11t + 10: A Comprehensive Guide
Quadratic equations are fundamental in algebra and mathematics, appearing in numerous applications from physics to economics. One such notable quadratic expression is 3t² + 11t + 10. This article explores the equation in depth, covering its significance, factoring, graph, real-world uses, and step-by-step solutions. Whether you're a student, teacher, or math enthusiast, this guide offers clear insight into working with — and understanding — this key quadratic.
Understanding the Context
What Is 3t² + 11t + 10?
The expression 3t² + 11t + 10 is a standard quadratic function in two variables, where:
- a = 3 (leading coefficient)
- b = 11 (middle coefficient)
- c = 10 (constant term)
It takes the general form:
at² + bt + c = 0
This quadratic equation can be used to model a wide range of real-world phenomena, such as projectile motion, revenue predictions, or optimization problems, making it essential in both theoretical and applied mathematics.
Key Insights
Factoring the Quadratic
Factoring helps simplify quadratic expressions, making them easier to solve or graph. Let’s factor 3t² + 11t + 10.
Step 1: Multiply a and c
3 × 10 = 30
Step 2: Find two numbers that multiply to 30 and add to b = 11
After testing:
5 × 6 = 30, and 5 + 6 = 11
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Perfect! These numbers will split the middle term.
Step 3: Rewrite the middle term
Break 11t into 5t + 6t:
3t² + 5t + 6t + 10
Step 4: Group and factor by pairing
(3t² + 5t) + (6t + 10)
t(3t + 5) + 2(3t + 5)
Now factor out the common binomial (3t + 5):
=(3t + 5)(t + 2)
Solving the Equation
To find the roots of 3t² + 11t + 10 = 0, set the factored form to zero:
(3t + 5)(t + 2) = 0
Set each factor equal to zero:
- 3t + 5 = 0 → t = –5/3
- t + 2 = 0 → t = –2
Roots: t = –5/3 and t = –2
