Solve The Equation: { (x-8)(2x+7)=0$}$The Solutions Are { X=$}$ { \square, \square$}$. (Use A Comma To Separate Answers As Needed.)

Mar 11, 2025 by ADMIN 133 views

**Solving Quadratic Equations: A Step-by-Step Guide**

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $(x-8)(2x+7)=0$ , and provide a step-by-step guide on how to find the solutions.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$ ) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Given Equation

The given equation is $(x-8)(2x+7)=0$ . This equation can be solved using the method of factoring, which involves expressing the equation as a product of two binomials.

Step 1: Factor the Equation

To factor the equation, we need to find two binomials whose product is equal to the given equation. In this case, we can factor the equation as follows:

$(x-8)(2x+7)=0$

This can be rewritten as:

$(x-8)(2x+7)=(x-8)(2)(x+\frac{7}{2})=0$

Step 2: Set Each Factor Equal to Zero

To find the solutions, we need to set each factor equal to zero and solve for $x$ . In this case, we have two factors:

$x-8=0$ and $2x+\frac{7}{2}=0$

Solving the First Factor

To solve the first factor, we need to isolate $x$ by adding $8$ to both sides of the equation:

$x-8+8=0+8$

This simplifies to:

$x=8$

Solving the Second Factor

To solve the second factor, we need to isolate $x$ by subtracting $\frac{7}{2}$ from both sides of the equation:

$2x+\frac{7}{2}-\frac{7}{2}=0-\frac{7}{2}$

This simplifies to:

$2x=-\frac{7}{2}$

Dividing both sides by $2$ gives:

$x=-\frac{7}{4}$

The Solutions

The solutions to the equation are $x=8$ and $x=-\frac{7}{4}$ . These values satisfy the original equation, and they can be verified by plugging them back into the equation.

Conclusion

Solving quadratic equations is an essential skill in mathematics, and it requires a step-by-step approach. In this article, we solved the equation $(x-8)(2x+7)=0$ using the method of factoring. We set each factor equal to zero and solved for $x$ , resulting in two solutions: $x=8$ and $x=-\frac{7}{4}$ . By following these steps, you can solve quadratic equations and gain a deeper understanding of this fundamental concept in mathematics.

Additional Tips and Resources

To solve quadratic equations, you can use the quadratic formula, which is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ .
You can also use graphing to solve quadratic equations. Graphing involves plotting the equation on a coordinate plane and finding the x-intercepts.
For more information on solving quadratic equations, you can refer to online resources such as Khan Academy, Mathway, or Wolfram Alpha.

Common Mistakes to Avoid

When solving quadratic equations, it's essential to set each factor equal to zero and solve for $x$ .
Make sure to simplify the equation and isolate $x$ before solving for its value.
Be careful when dividing or multiplying fractions, as this can lead to errors.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will address some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you better understand and solve these equations.

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 (in this case, $x$ ) is 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?

A: There are several methods to solve quadratic equations, including:

Factoring: This involves expressing the equation as a product of two binomials.
Quadratic formula: This involves using the formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to find the solutions.
Graphing: This involves plotting the equation on a coordinate plane and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to find the solutions to a quadratic equation. 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:

Identify the values of $a$ , $b$ , and $c$ in the quadratic equation.
Plug these values into the quadratic formula.
Simplify the expression to find the solutions.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared term, while a linear equation does not.

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

A: Yes, the quadratic formula can be used to solve all quadratic equations. However, it's worth noting that factoring and graphing can also be used to solve quadratic equations, and these methods may be more efficient or easier to use in certain cases.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not setting each factor equal to zero and solving for $x$ .
Not simplifying the equation and isolating $x$ before solving for its value.
Not being careful when dividing or multiplying fractions.

Q: How do I check my solutions to a quadratic equation?

A: To check your solutions to a quadratic equation, you can plug the solutions back into the original equation and verify that they satisfy the equation. You can also use the quadratic formula to check your solutions.

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

A: Quadratic equations have numerous real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

By understanding and solving quadratic equations, you can gain a deeper appreciation for the mathematical concepts that underlie many real-world phenomena.

Additional Resources

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

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding and solving quadratic equations, you can gain a deeper appreciation for the mathematical concepts that underlie many real-world phenomena. We hope this Q&A article has provided you with a comprehensive guide to help you better understand and solve quadratic equations.