Simplify The Expression: $ T^2 + 6t - 16 $

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Introduction

In mathematics, simplifying expressions is a crucial step in solving equations and inequalities. It involves rewriting an expression in a more compact and manageable form, often by combining like terms or factoring out common factors. In this article, we will focus on simplifying the quadratic expression $ t^2 + 6t - 16 $.

Understanding the Expression

The given expression is a quadratic expression in the variable $ t $. It consists of three terms: $ t^2 $, $ 6t $, and $ -16 $. The first term, $ t^2 $, is a squared term, while the second term, $ 6t $, is a linear term. The third term, $ -16 $, is a constant term.

Simplifying the Expression

To simplify the expression, we can start by factoring out the greatest common factor (GCF) of the three terms. However, in this case, there is no common factor that can be factored out. Therefore, we will focus on combining like terms.

Combining Like Terms

Like terms are terms that have the same variable and exponent. In this expression, the only like terms are the linear term $ 6t $ and the constant term $ -16 $. We can combine these two terms by adding or subtracting their coefficients.

Factoring the Expression

Since we cannot factor out a common factor, we can try to factor the expression by grouping. However, in this case, the expression cannot be factored by grouping.

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. However, in this case, we are not solving an equation, but rather simplifying an expression. Nevertheless, we can use the quadratic formula to find the roots of the quadratic expression.

Finding the Roots

The quadratic formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $ a = 1 $, $ b = 6 $, and $ c = -16 $. Plugging these values into the formula, we get:

t=−6±62−4(1)(−16)2(1)t = \frac{-6 \pm \sqrt{6^2 - 4(1)(-16)}}{2(1)}

Simplifying the expression under the square root, we get:

t=−6±36+642t = \frac{-6 \pm \sqrt{36 + 64}}{2}

t=−6±1002t = \frac{-6 \pm \sqrt{100}}{2}

t=−6±102t = \frac{-6 \pm 10}{2}

Therefore, the roots of the quadratic expression are:

t=−6+102=2t = \frac{-6 + 10}{2} = 2

t=−6−102=−8t = \frac{-6 - 10}{2} = -8

Conclusion

In this article, we simplified the quadratic expression $ t^2 + 6t - 16 $ by combining like terms and factoring the expression. We also used the quadratic formula to find the roots of the expression. The simplified expression is $ (t - 2)(t + 8) $.

Final Answer

The final answer is: (t−2)(t+8)\boxed{(t - 2)(t + 8)}

Introduction

In our previous article, we simplified the quadratic expression $ t^2 + 6t - 16 $ by combining like terms and factoring the expression. We also used the quadratic formula to find the roots of the expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying the expression.

Q&A

Q: What is the greatest common factor (GCF) of the three terms in the expression?

A: The GCF of the three terms in the expression is 1, since there is no common factor that can be factored out.

Q: Can the expression be factored by grouping?

A: No, the expression cannot be factored by grouping.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to find the roots of the expression?

A: To use the quadratic formula, you need to plug in the values of $ a $, $ b $, and $ c $ into the formula. In this case, $ a = 1 $, $ b = 6 $, and $ c = -16 $. Plugging these values into the formula, you get:

t=−6±62−4(1)(−16)2(1)t = \frac{-6 \pm \sqrt{6^2 - 4(1)(-16)}}{2(1)}

Simplifying the expression under the square root, you get:

t=−6±36+642t = \frac{-6 \pm \sqrt{36 + 64}}{2}

t=−6±1002t = \frac{-6 \pm \sqrt{100}}{2}

t=−6±102t = \frac{-6 \pm 10}{2}

Therefore, the roots of the quadratic expression are:

t=−6+102=2t = \frac{-6 + 10}{2} = 2

t=−6−102=−8t = \frac{-6 - 10}{2} = -8

Q: What is the final answer to the expression?

A: The final answer to the expression is $ (t - 2)(t + 8) $.

Q: Can I use the quadratic formula to solve quadratic equations?

A: Yes, the quadratic formula can be used to solve quadratic equations. It is a powerful tool for finding the roots of quadratic equations.

Q: What is the difference between a quadratic expression and a quadratic equation?

A: A quadratic expression is an algebraic expression that contains a squared variable, while a quadratic equation is an equation that contains a squared variable and is set equal to zero.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying the expression $ t^2 + 6t - 16 $. We also provided some additional information about the quadratic formula and quadratic equations.

Final Answer

The final answer is: (t−2)(t+8)\boxed{(t - 2)(t + 8)}