What Are The Zeros Of The Equation Shown Below?$(2x+3)(x-5)=0$A. $x=-1.5, X=5$ B. $x=0.67, X=5$ C. $x=-0.67, X=-5$ D. $x=1.5, X=-5$

Feb 28, 2025 by ADMIN 137 views

**Solving the Equation: Finding the Zeros of a Quadratic Expression**

Introduction

In algebra, a zero of an equation is a value of the variable that makes the equation true. In other words, it is a solution to the equation. In this article, we will focus on finding the zeros of a quadratic equation in the form of $(ax+b)(cx+d)=0$ . We will use the given equation $(2x+3)(x-5)=0$ as an example to demonstrate the steps involved in solving this type of equation.

Understanding the Equation

The given equation is a quadratic equation in the form of $(ax+b)(cx+d)=0$ . To find the zeros of this equation, we need to set each factor equal to zero and solve for the variable $x$ . The equation can be rewritten as:

(2x+3)(x-5)=0

Step 1: Setting Each Factor Equal to Zero

To find the zeros of the equation, we need to set each factor equal to zero and solve for the variable $x$ . We can do this by using the following steps:

Set the first factor equal to zero: $2x+3=0$
Set the second factor equal to zero: $x-5=0$

Solving the First Factor

To solve the first factor, we need to isolate the variable $x$ . We can do this by subtracting 3 from both sides of the equation and then dividing both sides by 2.

2x+3=0

Subtracting 3 from both sides:

2x=-3

Dividing both sides by 2:

x=-\frac{3}{2}

Solving the Second Factor

To solve the second factor, we need to isolate the variable $x$ . We can do this by adding 5 to both sides of the equation.

x-5=0

Adding 5 to both sides:

x=5

Finding the Zeros of the Equation

Now that we have solved both factors, we can find the zeros of the equation by substituting the values of $x$ back into the original equation.

Substituting $x=-\frac{3}{2}$ into the original equation: $(2(-\frac{3}{2})+3)(-\frac{3}{2}-5)=0$ Simplifying the equation: $(-3+3)(-\frac{3}{2}-5)=0$ Further simplifying the equation: $0=0$
Substituting $x=5$ into the original equation: $(2(5)+3)(5-5)=0$ Simplifying the equation: $(10+3)(0)=0$ Further simplifying the equation: $13(0)=0$

Conclusion

In conclusion, the zeros of the equation $(2x+3)(x-5)=0$ are $x=-\frac{3}{2}$ and $x=5$ . These values make the equation true, and they are the solutions to the equation.

Answer

The correct answer is:

A. $x=-1.5, x=5$

Introduction

In our previous article, we discussed how to solve quadratic equations in the form of $(ax+b)(cx+d)=0$ . We used the equation $(2x+3)(x-5)=0$ as an example to demonstrate the steps involved in solving this type of equation. In this article, we will answer some frequently asked questions related to solving quadratic equations.

Q: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It can be written in the form of $ax^2+bx+c=0$ or $(ax+b)(cx+d)=0$ .

Q: How do I solve a quadratic equation?

To solve a quadratic equation, you need to follow these steps:

Set each factor equal to zero and solve for the variable.
Use the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ .
Use factoring: If the equation can be factored, set each factor equal to zero and solve for the variable.

Q: What is the quadratic formula?

The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Q: When should I use the quadratic formula?

You should use the quadratic formula when:

The equation cannot be factored.
The equation is in the form of $ax^2+bx+c=0$ .
You are given the coefficients $a$ , $b$ , and $c$ .

Q: What is the difference between the quadratic formula and factoring?

The quadratic formula and factoring are two different methods of solving quadratic equations. The quadratic formula is a formula that can be used to solve quadratic equations, while factoring involves setting each factor equal to zero and solving for the variable.

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

Yes, you can use the quadratic formula to solve equations with complex solutions. The quadratic formula will give you two solutions, one of which may be complex.

Q: How do I determine if a quadratic equation has real or complex solutions?

To determine if a quadratic equation has real or complex solutions, you need to look at the discriminant, which is given by $b^2-4ac$ . If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: What is the discriminant?

The discriminant is a value that can be used to determine the nature of the solutions of a quadratic equation. It is given by $b^2-4ac$ .

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

Yes, you can use the quadratic formula to solve equations with rational solutions. The quadratic formula will give you two solutions, one of which may be rational.

Conclusion

In conclusion, solving quadratic equations is an important topic in algebra. By understanding the different methods of solving quadratic equations, such as the quadratic formula and factoring, you can solve a wide range of equations. We hope this article has been helpful in answering some of the frequently asked questions related to solving quadratic equations.

Additional Resources

If you are looking for additional resources to help you learn more about solving quadratic equations, here are a few suggestions:

Khan Academy: Quadratic Equations
Mathway: Quadratic Equations
Wolfram Alpha: Quadratic Equations

Practice Problems

Here are a few practice problems to help you practice solving quadratic equations:

Solve the equation $x^2+5x+6=0$ using the quadratic formula.
Solve the equation $(x+2)(x-3)=0$ using factoring.
Solve the equation $x^2-7x+12=0$ using the quadratic formula.

We hope these practice problems are helpful in helping you practice solving quadratic equations.