Solve The Equation 8 X 2 − 14 X + 5 = 0 8x^2 - 14x + 5 = 0 8 X 2 − 14 X + 5 = 0 .

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. 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.

In this article, we will focus on solving the quadratic equation $8x^2 - 14x + 5 = 0$. We will use various methods to find the roots of this equation, including factoring, the quadratic formula, and completing the square.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. In our case, $a = 8$, $b = -14$, and $c = 5$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(8)(5)}}{2(8)}$$

Simplifying the expression, we get:

$$x = \frac{14 \pm \sqrt{196 - 160}}{16}$$

$$x = \frac{14 \pm \sqrt{36}}{16}$$

$$x = \frac{14 \pm 6}{16}$$

Solving for the Roots

Now that we have the quadratic formula, we can solve for the roots of the equation. We have two possible solutions:

$$x = \frac{14 + 6}{16}$$

$$x = \frac{20}{16}$$

$$x = \frac{5}{4}$$

$$x = \frac{14 - 6}{16}$$

$$x = \frac{8}{16}$$

$$x = \frac{1}{2}$$

Factoring the Quadratic Equation

Another method for solving quadratic equations is factoring. We can try to factor the quadratic equation $8x^2 - 14x + 5 = 0$ by finding two numbers whose product is $8 \times 5 = 40$ and whose sum is $-14$. These numbers are $-10$ and $-4$, since $-10 \times -4 = 40$ and $-10 + (-4) = -14$. Therefore, we can write the quadratic equation as:

$$8x^2 - 14x + 5 = (4x - 5)(2x - 1) = 0$$

Solving for the Roots using Factoring

Now that we have factored the quadratic equation, we can solve for the roots by setting each factor equal to zero:

$$4x - 5 = 0$$

$$2x - 1 = 0$$

Solving for $x$, we get:

$$x = \frac{5}{4}$$

$$x = \frac{1}{2}$$

Completing the Square

Another method for solving quadratic equations is completing the square. We can rewrite the quadratic equation $8x^2 - 14x + 5 = 0$ as:

$$8x^2 - 14x = -5$$

$$8x^2 - 14x + \left(\frac{7}{4}\right)^2 = -5 + \left(\frac{7}{4}\right)^2$$

$$8x^2 - 14x + \frac{49}{16} = -5 + \frac{49}{16}$$

$$8x^2 - 14x + \frac{49}{16} = \frac{-80 + 49}{16}$$

$$8x^2 - 14x + \frac{49}{16} = \frac{-31}{16}$$

Solving for the Roots using Completing the Square

Now that we have completed the square, we can solve for the roots by taking the square root of both sides:

$$\left(x - \frac{7}{8}\right)^2 = \frac{-31}{16}$$

$$x - \frac{7}{8} = \pm \sqrt{\frac{-31}{16}}$$

$$x = \frac{7}{8} \pm \sqrt{\frac{-31}{16}}$$

Conclusion

In this article, we have solved the quadratic equation $8x^2 - 14x + 5 = 0$ using various methods, including the quadratic formula, factoring, and completing the square. We have found two roots, $x = \frac{5}{4}$ and $x = \frac{1}{2}$. These roots are the solutions to the quadratic equation, and they can be used to solve a wide range of problems in mathematics and other fields.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. Some examples of applications of quadratic equations include:

• Projectile Motion: Quadratic equations can be used to model the trajectory of a projectile, such as a thrown ball or a launched rocket.
• Optimization: Quadratic equations can be used to optimize functions, such as finding the maximum or minimum value of a function.
• Signal Processing: Quadratic equations can be used to filter signals and remove noise.
• Computer Graphics: Quadratic equations can be used to create 3D models and animations.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we have solved the quadratic equation $8x^2 - 14x + 5 = 0$ using various methods, including the quadratic formula, factoring, and completing the square. We have found two roots, $x = \frac{5}{4}$ and $x = \frac{1}{2}$. These roots are the solutions to the quadratic equation, and they can be used to solve a wide range of problems in mathematics and other fields.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we will answer some of the most frequently asked questions about quadratic equations, including their definition, methods of solution, and applications.

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 methods for solving quadratic equations, including the quadratic formula, factoring, and completing the square. The quadratic formula is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I factor a quadratic equation?

A: Factoring a quadratic equation involves finding two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the factors of the quadratic equation. Once you have found the factors, you can set each factor equal to zero and solve for $x$.

Q: What is completing the square?

A: Completing the square is a method for solving quadratic equations by rewriting the equation in a form that allows you to easily solve for $x$. It involves adding and subtracting a constant term to the equation, which allows you to create a perfect square trinomial.

Q: How do I apply quadratic equations in real-life situations?

A: Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. Some examples of applications of quadratic equations include:

• Projectile Motion: Quadratic equations can be used to model the trajectory of a projectile, such as a thrown ball or a launched rocket.
• Optimization: Quadratic equations can be used to optimize functions, such as finding the maximum or minimum value of a function.
• Signal Processing: Quadratic equations can be used to filter signals and remove noise.
• Computer Graphics: Quadratic equations can be used to create 3D models and animations.

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

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

• Not checking the discriminant: The discriminant is the expression under the square root in the quadratic formula. If the discriminant is negative, the equation has no real solutions.
• Not simplifying the expression: Make sure to simplify the expression under the square root before solving for $x$.
• Not checking for extraneous solutions: Make sure to check for extraneous solutions, which are solutions that do not satisfy the original equation.

Q: How do I choose the best method for solving a quadratic equation?

A: The best method for solving a quadratic equation depends on the specific equation and the desired solution. Some methods, such as the quadratic formula, are more general and can be used to solve any quadratic equation. Other methods, such as factoring, may be more efficient for certain types of equations.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we have answered some of the most frequently asked questions about quadratic equations, including their definition, methods of solution, and applications. We hope that this article has provided you with a better understanding of quadratic equations and how to apply them in real-life situations.