Given The Function $g(t) = T^2 - T - 42$:1) What Are The Zeros Of The Function?Write The Smaller $t$ First, And The Larger $t$ Second.- Smaller $t =$ $\square$- Larger $t =$ $\square$2)

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Introduction

In mathematics, 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 ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable. In this article, we will focus on solving a specific quadratic equation, g(t)=t2−t−42g(t) = t^2 - t - 42, to find its zeros.

What are Zeros of a Function?

The zeros of a function are the values of the variable that make the function equal to zero. In other words, they are the solutions to the equation f(x)=0f(x) = 0. For a quadratic equation, the zeros can be found using various methods, including factoring, the quadratic formula, and graphing.

Method 1: Factoring

To find the zeros of the function g(t)=t2−t−42g(t) = t^2 - t - 42, we can try to factor the quadratic expression. Factoring involves expressing the quadratic expression as a product of two binomials. In this case, we can factor the expression as follows:

t2−t−42=(t−7)(t+6)t^2 - t - 42 = (t - 7)(t + 6)

Finding the Zeros

Now that we have factored the quadratic expression, we can find the zeros by setting each factor equal to zero and solving for tt. Let's start with the first factor, t−7=0t - 7 = 0.

t−7=0t - 7 = 0

t=7t = 7

This is the first zero of the function. Now, let's move on to the second factor, t+6=0t + 6 = 0.

t+6=0t + 6 = 0

t=−6t = -6

This is the second zero of the function.

Conclusion

In this article, we have solved the quadratic equation g(t)=t2−t−42g(t) = t^2 - t - 42 to find its zeros. We used the method of factoring to express the quadratic expression as a product of two binomials and then set each factor equal to zero to find the zeros. The smaller zero is t=−6t = -6, and the larger zero is t=7t = 7.

Discussion

The zeros of a function are an important concept in mathematics, and they have many real-world applications. For example, in physics, the zeros of a function can represent the equilibrium points of a system, while in engineering, they can represent the critical points of a system.

Real-World Applications

The concept of zeros of a function has many real-world applications. For example:

  • Physics: In physics, the zeros of a function can represent the equilibrium points of a system. For instance, the potential energy of a simple harmonic oscillator is given by the function U(x)=12kx2U(x) = \frac{1}{2}kx^2, where kk is the spring constant and xx is the displacement from the equilibrium position. The zeros of this function represent the equilibrium points of the system.
  • Engineering: In engineering, the zeros of a function can represent the critical points of a system. For instance, the stress on a beam is given by the function S(x)=MxIS(x) = \frac{Mx}{I}, where MM is the moment, xx is the distance from the support, and II is the moment of inertia. The zeros of this function represent the critical points of the system.

Conclusion

In conclusion, the zeros of a function are an important concept in mathematics, and they have many real-world applications. By understanding how to find the zeros of a function, we can gain a deeper understanding of the behavior of systems and make more accurate predictions about their behavior.

Final Answer

The final answer is:

  • Smaller t=t = -6
  • Larger t=t = 7
    Quadratic Equations: A Q&A Guide =====================================

Introduction

In our previous article, we explored the concept of zeros of a function and how to find them using the method of factoring. In this article, we will continue to delve deeper into the world of quadratic equations and answer some frequently asked questions.

Q&A

Q: What is a quadratic equation?

A: 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 ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable.

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, try to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to find the solutions to a quadratic equation. The quadratic formula is given by:

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

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when the quadratic equation cannot be factored. The quadratic formula can be used to find the solutions to any quadratic equation, regardless of whether it can be factored or not.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It is given by b2−4acb^2 - 4ac. The discriminant can be used to determine the nature of the solutions to a quadratic equation.

Q: What does the discriminant tell us about the solutions?

A: The discriminant can tell us whether the solutions to a quadratic equation are real or complex. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, first find the zeros of the equation by using the quadratic formula or factoring. Then, use the zeros to plot the x-intercepts of the graph. Finally, use the vertex of the parabola to determine the direction of the graph.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the graph that is lowest or highest point. The vertex can be found using the formula x=−b2ax = -\frac{b}{2a}.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, first find the x-coordinate of the vertex using the formula x=−b2ax = -\frac{b}{2a}. Then, substitute this value into the equation to find the y-coordinate of the vertex.

Q: What is the axis of symmetry?

A: The axis of symmetry is a line that passes through the vertex of a parabola and is perpendicular to the x-axis. The axis of symmetry can be used to determine the direction of the graph.

Q: How do I find the axis of symmetry?

A: To find the axis of symmetry, first find the vertex of the parabola. Then, use the formula x=−b2ax = -\frac{b}{2a} to find the x-coordinate of the axis of symmetry.

Conclusion

In conclusion, quadratic equations are an important concept in mathematics, and they have many real-world applications. By understanding how to solve quadratic equations, we can gain a deeper understanding of the behavior of systems and make more accurate predictions about their behavior.

Final Answer

The final answer is:

  • The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • The discriminant is: b2−4acb^2 - 4ac
  • The vertex of a parabola is: x=−b2ax = -\frac{b}{2a}
  • The axis of symmetry is: x=−b2ax = -\frac{b}{2a}