Solve The Equation { (3x - 5)(x + 2) = 0$}$.

Introduction

In mathematics, equations are a fundamental concept that helps us understand the relationship between variables. Solving equations is a crucial skill that is used in various fields, including physics, engineering, and economics. In this article, we will focus on solving a quadratic equation of the form $(3x - 5)(x + 2) = 0$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is a quadratic equation, which is a polynomial equation of degree two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this case, the equation is $(3x - 5)(x + 2) = 0$. To solve this equation, we need to find the values of $x$ that make the equation true.

Step 1: Expand the Equation

The first step in solving the equation is to expand it using the distributive property. This means that we need to multiply each term in the first parentheses by each term in the second parentheses.

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

Expanding the equation, we get:

3x(x + 2) - 5(x + 2) = 3x^2 + 6x - 5x - 10

Simplifying the equation, we get:

3x^2 + 6x - 5x - 10 = 3x^2 + x - 10

Step 2: Set Each Factor Equal to Zero

The next step is to set each factor equal to zero. This means that we need to set the first factor, $3x - 5$, equal to zero and the second factor, $x + 2$, equal to zero.

3x - 5 = 0
x + 2 = 0

Step 3: Solve for x

Now that we have set each factor equal to zero, we can solve for $x$. To solve for $x$, we need to isolate $x$ on one side of the equation.

For the first equation, $3x - 5 = 0$, we can add 5 to both sides of the equation to get:

3x - 5 + 5 = 0 + 5
3x = 5

Dividing both sides of the equation by 3, we get:

3x/3 = 5/3
x = 5/3

For the second equation, $x + 2 = 0$, we can subtract 2 from both sides of the equation to get:

x + 2 - 2 = 0 - 2
x = -2

Conclusion

In this article, we have solved the quadratic equation $(3x - 5)(x + 2) = 0$ using the steps outlined above. We expanded the equation, set each factor equal to zero, and solved for $x$. The solutions to the equation are $x = 5/3$ and $x = -2$. These solutions represent the values of $x$ that make the equation true.

Real-World Applications

Solving quadratic equations has many real-world applications. For example, in physics, quadratic equations are used to model the motion of objects under the influence of gravity. In engineering, quadratic equations are used to design bridges and buildings. In economics, quadratic equations are used to model the behavior of markets and economies.

Tips and Tricks

When solving quadratic equations, it is essential to follow the steps outlined above. Here are some tips and tricks to help you solve quadratic equations:

• Make sure to expand the equation correctly using the distributive property.
• Set each factor equal to zero and solve for $x$.
• Check your solutions by plugging them back into the original equation.
• Use the quadratic formula to solve quadratic equations when the equation is in the form $ax^2 + bx + c = 0$.

Conclusion

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Introduction

In our previous article, we solved the quadratic equation $(3x - 5)(x + 2) = 0$ using the steps outlined above. We expanded the equation, set each factor equal to zero, and solved for $x$. The solutions to the equation are $x = 5/3$ and $x = -2$. These solutions represent the values of $x$ that make the equation true. In this article, we will answer some frequently asked questions about solving quadratic equations.

Q: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

To solve a quadratic equation, you need to follow the steps outlined above:

1. Expand the equation using the distributive property.
2. Set each factor equal to zero and solve for $x$.
3. Check your solutions by plugging them back into the original equation.

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

A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is $ax + b = 0$, where $a$ and $b$ are constants.

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

Yes, you can use the quadratic formula to solve quadratic equations when the equation is in the form $ax^2 + bx + c = 0$. The quadratic formula is:

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

Q: What is the quadratic formula?

The quadratic formula is a formula that is used to solve quadratic equations. It is:

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

Q: How do I use the quadratic formula?

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

Q: What are some real-world applications of quadratic equations?

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

• Modeling the motion of objects under the influence of gravity in physics.
• Designing bridges and buildings in engineering.
• Modeling the behavior of markets and economies in economics.

Q: What are some tips and tricks for solving quadratic equations?

Here are some tips and tricks for solving quadratic equations:

• Make sure to expand the equation correctly using the distributive property.
• Set each factor equal to zero and solve for $x$.
• Check your solutions by plugging them back into the original equation.
• Use the quadratic formula to solve quadratic equations when the equation is in the form $ax^2 + bx + c = 0$.

Conclusion

Solving quadratic equations is a crucial skill that is used in various fields. In this article, we have answered some frequently asked questions about solving quadratic equations. We have also provided some tips and tricks for solving quadratic equations. By following the steps outlined above and using the quadratic formula, you can solve quadratic equations with ease.

Additional Resources

If you want to learn more about solving quadratic equations, here are some additional resources:

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

Conclusion

Solving quadratic equations is a crucial skill that is used in various fields. In this article, we have answered some frequently asked questions about solving quadratic equations. We have also provided some tips and tricks for solving quadratic equations. By following the steps outlined above and using the quadratic formula, you can solve quadratic equations with ease.