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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.
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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
