Solve For $t$:$3t^2 + 84t = 0$Write Each Solution As An Integer, Proper Fraction, Or Improper Fraction. If There Are Multiple Solutions, Separate Them With Commas.\$t = \square$[/tex\]

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

In this article, we will focus on solving a specific quadratic equation, $3t^2 + 84t = 0$. Our goal is to find the value of $t$ that satisfies this equation. To do this, we will use various techniques and strategies to isolate the variable $t$.

Factoring the Quadratic Equation

One of the most common methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. In this case, we can factor out the greatest common factor (GCF) of the two terms, which is $3t$.

3t2+84t=03t^2 + 84t = 0

3t(t+28)=03t(t + 28) = 0

Now we have factored the quadratic equation into two binomials. To find the value of $t$, we need to set each binomial equal to zero and solve for $t$.

Setting Each Binomial Equal to Zero

We have two binomials: $3t$ and $(t + 28)$. We will set each binomial equal to zero and solve for $t$.

Solving for $3t = 0$

3t=03t = 0

t=0t = 0

Solving for $(t + 28) = 0$

(t+28)=0(t + 28) = 0

t+28=0t + 28 = 0

t=−28t = -28

Conclusion

In this article, we have solved the quadratic equation $3t^2 + 84t = 0$ using the factoring method. We factored the quadratic equation into two binomials and set each binomial equal to zero to find the value of $t$. The solutions to the equation are $t = 0$ and $t = -28$. These solutions can be expressed as integers, proper fractions, or improper fractions.

Final Answer

t = 0, -28$<br/> # Solve for $t$: $3t^2 + 84t = 0$ - Q&A ## Introduction In our previous article, we solved the quadratic equation $3t^2 + 84t = 0$ using the factoring method. We factored the quadratic equation into two binomials and set each binomial equal to zero to find the value of $t$. In this article, we will provide a Q&A section to address any questions or concerns that readers may have. ## Q&A ### Q: What is the difference between a quadratic equation and a linear equation? A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. A linear equation, on the other hand, is a polynomial equation of degree one, which means the highest power of the variable is one. ### Q: How do I know if a quadratic equation can be factored? A: A quadratic equation can be factored if it can be expressed as a product of two binomials. To determine if a quadratic equation can be factored, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term. ### Q: What is the greatest common factor (GCF) of two terms? A: The greatest common factor (GCF) of two terms is the largest number that divides both terms evenly. In the case of the quadratic equation $3t^2 + 84t = 0$, the GCF of the two terms is $3t$. ### Q: How do I set each binomial equal to zero to find the value of $t$? A: To set each binomial equal to zero, simply set the first term of each binomial equal to zero and solve for $t$. For example, in the case of the quadratic equation $3t^2 + 84t = 0$, we set $3t = 0$ and $t + 28 = 0$ to find the value of $t$. ### Q: What are the solutions to the quadratic equation $3t^2 + 84t = 0$? A: The solutions to the quadratic equation $3t^2 + 84t = 0$ are $t = 0$ and $t = -28$. ### Q: Can the solutions to the quadratic equation $3t^2 + 84t = 0$ be expressed as integers, proper fractions, or improper fractions? A: Yes, the solutions to the quadratic equation $3t^2 + 84t = 0$ can be expressed as integers, proper fractions, or improper fractions. The solution $t = 0$ can be expressed as an integer, while the solution $t = -28$ can be expressed as an integer or a proper fraction. ## Conclusion In this article, we have provided a Q&A section to address any questions or concerns that readers may have about solving the quadratic equation $3t^2 + 84t = 0$. We have covered topics such as the difference between quadratic and linear equations, factoring, greatest common factors, and setting binomials equal to zero. We hope that this Q&A section has been helpful in clarifying any confusion and providing a better understanding of the material. ## Final Answer $t = 0, -28