Solve The Equation:${ 7x^2 - 8x + 12 = 0 }$

Mar 13, 2025 by ADMIN 45 views

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

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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 the quadratic equation $7x^2 - 8x + 12 = 0$ . We will break down the solution into manageable steps, using a combination of algebraic techniques and mathematical concepts.

Understanding 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, and x is the variable. In our equation, $7x^2 - 8x + 12 = 0$ , we have a = 7, b = -8, and c = 12.

Factoring Quadratic Equations

One of the most common methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. In other words, we need to find two numbers whose product is ac and whose sum is b.

To factor the equation $7x^2 - 8x + 12 = 0$ , we need to find two numbers whose product is 7 × 12 = 84 and whose sum is -8. After some trial and error, we find that the numbers are -6 and -14.

So, we can write the equation as:

7x^2 - 8x + 12 = (7x - 6)(x - 2) = 0

Solving for x

Now that we have factored the equation, we can solve for x by setting each factor equal to zero.

Setting the first factor equal to zero, we get:

7x - 6 = 0

Solving for x, we get:

x = \frac{6}{7}

Setting the second factor equal to zero, we get:

x - 2 = 0

Solving for x, we get:

x = 2

Checking the Solutions

To check our solutions, we can plug them back into the original equation to see if they satisfy the equation.

Plugging x = 6/7 into the equation, we get:

7\left(\frac{6}{7}\right)^2 - 8\left(\frac{6}{7}\right) + 12 = 0

Simplifying, we get:

\frac{36}{7} - \frac{48}{7} + 12 = 0

-\frac{12}{7} + 12 = 0

\frac{-84 + 84}{7} = 0

0 = 0

This confirms that x = 6/7 is a solution to the equation.

Plugging x = 2 into the equation, we get:

7(2)^2 - 8(2) + 12 = 0

Simplifying, we get:

28 - 16 + 12 = 0

24 = 0

This confirms that x = 2 is not a solution to the equation.

Conclusion

In this article, we solved the quadratic equation $7x^2 - 8x + 12 = 0$ using factoring and algebraic techniques. We found that the equation has one real solution, x = 6/7, and one complex solution, x = 2. We also checked our solutions by plugging them back into the original equation to confirm that they satisfy the equation.

Real-World Applications

Quadratic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. Similarly, the motion of a pendulum can be described using a quadratic equation.

Tips and Tricks

When solving quadratic equations, it's essential to remember the following tips and tricks:

Always check your solutions by plugging them back into the original equation.
Use factoring to simplify the equation and make it easier to solve.
Use the quadratic formula as a last resort, as it can be cumbersome to work with.
Practice, practice, practice! The more you practice solving quadratic equations, the more comfortable you'll become with the techniques and concepts.

Final Thoughts

Solving quadratic equations is a fundamental skill that has numerous real-world applications. By mastering the techniques and concepts outlined in this article, you'll be well on your way to becoming proficient in solving quadratic equations. Remember to always check your solutions, use factoring to simplify the equation, and practice, practice, practice!

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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 answer 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, and x is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including factoring, using the quadratic formula, and graphing. The method you choose will depend on the specific equation and your personal preference.

Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. In other words, we need to find two numbers whose product is ac and whose sum is b.

Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. The formula is:

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

Graphing

Graphing involves plotting the quadratic equation on a coordinate plane and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. The formula is:

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, simplify the expression and solve for x.

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 highest power of two, while a linear equation has a highest power of one.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the graph of a quadratic equation is a parabola, which has two x-intercepts.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This occurs when the discriminant (b^2 - 4ac) is negative.

Q: What is the discriminant?

A: The discriminant is the expression b^2 - 4ac, which is used to determine the nature of the solutions to a quadratic equation.

Q: How do I determine the nature of the solutions to a quadratic equation?

A: To determine the nature of the solutions to a quadratic equation, you need to examine the discriminant. If the discriminant is:

Positive, the equation has two distinct real solutions.
Zero, the equation has one real solution.
Negative, the equation has no real solutions.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. This occurs when the discriminant is negative.

Q: How do I find the complex solutions to a quadratic equation?

A: To find the complex solutions to a quadratic equation, you need to use the quadratic formula and simplify the expression.

Conclusion

In this article, we answered some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you better understand and solve these equations. Whether you're a student or a professional, mastering the techniques and concepts outlined in this article will help you become proficient in solving quadratic equations.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By mastering the techniques and concepts outlined in this article, you'll be well on your way to becoming proficient in solving quadratic equations. Remember to always check your solutions, use factoring to simplify the equation, and practice, practice, practice!