Given The Function $g(x)=x^2+3x-4$:1) What Are The Zeros Of The Function?- Write The Smaller $x$ First, And The Larger \$x$[/tex\] Second. - Smaller $x=$ $\square$ - Larger $x=$

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 ${ax^2 + bx + c = 0}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable. One of the most important concepts in quadratic equations is finding the zeros of the function, which are the values of ${x}$ that make the function equal to zero. In this article, we will focus on solving quadratic equations and finding the zeros of a given function.

The Function $g(x)=x^2+3x-4$

The given function is $g(x)=x^2+3x-4$. To find the zeros of this function, we need to set the function equal to zero and solve for ${x}$. This can be done by using the quadratic formula or by factoring the quadratic expression.

Factoring the Quadratic Expression

To factor the quadratic expression, we need to find two numbers whose product is ${-4}$ and whose sum is ${3}$. These numbers are ${4}$ and ${-1}$, because ${4 \times (-1) = -4}$ and ${4 + (-1) = 3}$. Therefore, we can write the quadratic expression as:

$$g(x) = (x + 4)(x - 1)$$

Setting the Function Equal to Zero

To find the zeros of the function, we need to set the function equal to zero and solve for ${x}$. This can be done by setting each factor equal to zero and solving for ${x}$.

$$(x + 4) = 0 \quad \text{or} \quad (x - 1) = 0$$

Solving for $x$

Solving for ${x}$ in each equation, we get:

$$x + 4 = 0 \quad \Rightarrow \quad x = -4$$

$$x - 1 = 0 \quad \Rightarrow \quad x = 1$$

The Zeros of the Function

Therefore, the zeros of the function $g(x)=x^2+3x-4$ are ${-4}$ and ${1}$. The smaller ${x}$ is ${-4}$, and the larger ${x}$ is ${1}$.

Conclusion

In this article, we have discussed how to solve quadratic equations and find the zeros of a given function. We have used the quadratic formula and factoring to solve the quadratic expression, and we have found the zeros of the function to be ${-4}$ and ${1}$. This is an important concept in mathematics, and it has many applications in science, engineering, and economics.

Applications of Quadratic Equations

Quadratic equations have many applications in science, engineering, and economics. Some examples include:

• Projectile Motion: Quadratic equations are used to model the trajectory of a projectile under the influence of gravity.
• Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
• Economics: Quadratic equations are used to model economic systems, such as supply and demand curves.
• Physics: Quadratic equations are used to model physical systems, such as the motion of a pendulum or a spring-mass system.

Real-World Examples

Quadratic equations have many real-world applications. Some examples include:

• Designing a Trajectory: Quadratic equations are used to design the trajectory of a spacecraft or a missile.
• Optimizing a System: Quadratic equations are used to optimize a system, such as finding the maximum or minimum value of a function.
• Modeling a Physical System: Quadratic equations are used to model a physical system, such as the motion of a pendulum or a spring-mass system.

Tips and Tricks

Here are some tips and tricks for solving quadratic equations:

• Use the Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations.
• Factor the Quadratic Expression: Factoring the quadratic expression can make it easier to solve the equation.
• Use a Calculator: A calculator can be a useful tool for solving quadratic equations.
• Check Your Work: Always check your work to make sure that you have found the correct solution.

Conclusion

In conclusion, quadratic equations are an important concept in mathematics, and they have many applications in science, engineering, and economics. By understanding how to solve quadratic equations and find the zeros of a function, we can apply this knowledge to real-world problems and make informed decisions.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have many applications in science, engineering, and economics. In our previous article, we discussed how to solve quadratic equations and find the zeros of a function. In this article, we will provide a Q&A guide to help you understand quadratic equations better.

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 ${ax^2 + bx + c = 0}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable.

Q: How Do I Solve a Quadratic Equation?

A: There are several ways to solve a quadratic equation, including:

• Factoring: If the quadratic expression can be factored, you can set each factor equal to zero and solve for ${x}$.
• Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the Quadratic Formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How Do I Use the Quadratic Formula?

A: To use the quadratic formula, you need to plug in the values of ${a}$, ${b}$, and ${c}$ into the formula. Then, you can simplify the expression and solve for ${x}$.

Q: What is the Difference Between a Zero and a Root?

A: A zero and a root are two different terms that are often used interchangeably. However, a zero is a value of ${x}$ that makes the function equal to zero, while a root is a value of ${x}$ that makes the function equal to zero or a multiple of the function.

Q: How Do I Find the Zeros of a Function?

A: To find the zeros of a function, you need to set the function equal to zero and solve for ${x}$. This can be done by using the quadratic formula or by factoring the quadratic expression.

Q: What is the Sign of the Discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. If the discriminant is positive, the quadratic equation has two distinct real roots. If the discriminant is zero, the quadratic equation has one real root. If the discriminant is negative, the quadratic equation has no real roots.

Q: How Do I Determine the Number of Real Roots?

A: To determine the number of real roots, you need to examine the discriminant. If the discriminant is positive, the quadratic equation has two distinct real roots. If the discriminant is zero, the quadratic equation has one real root. If the discriminant is negative, the quadratic equation has no real roots.

Q: What is the Relationship Between the Coefficients and the Roots?

A: The coefficients of the quadratic equation are related to the roots of the equation. Specifically, the sum of the roots is equal to ${-\frac{b}{a}}$ and the product of the roots is equal to ${\frac{c}{a}}$.

Q: How Do I Use the Relationship Between the Coefficients and the Roots?

A: To use the relationship between the coefficients and the roots, you need to plug in the values of the coefficients into the formulas and solve for the roots.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have many applications in science, engineering, and economics. By understanding how to solve quadratic equations and find the zeros of a function, you can apply this knowledge to real-world problems and make informed decisions. We hope that this Q&A guide has helped you to better understand quadratic equations and how to solve them.