Solve The Equation For $x$:$\[x^2 - 4x - 32 = 0\\]

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as 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 quadratic equations of the form $x^2 - 4x - 32 = 0$.

Understanding 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. To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the given equation.

Identifying the Coefficients

In the given equation $x^2 - 4x - 32 = 0$, we can identify the coefficients as follows:

• $a = 1$ (coefficient of $x^2$)
• $b = -4$ (coefficient of $x$)
• $c = -32$ (constant term)

Applying the Quadratic Formula

Now that we have identified the coefficients, we can apply the quadratic formula to solve for $x$. Plugging in the values of $a$, $b$, and $c$ into the quadratic formula, we get:

$$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-32)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{4 \pm \sqrt{16 + 128}}{2}$$

$$x = \frac{4 \pm \sqrt{144}}{2}$$

$$x = \frac{4 \pm 12}{2}$$

Solving for $x$

Now we have two possible solutions for $x$:

$$x = \frac{4 + 12}{2} = 8$$

$$x = \frac{4 - 12}{2} = -4$$

Therefore, the solutions to the equation $x^2 - 4x - 32 = 0$ are $x = 8$ and $x = -4$.

Checking the Solutions

To verify that these solutions are correct, we can plug them back into the original equation:

$$8^2 - 4(8) - 32 = 64 - 32 - 32 = 0$$

$$(-4)^2 - 4(-4) - 32 = 16 + 16 - 32 = 0$$

Both solutions satisfy the original equation, so we can be confident that our solutions are correct.

Conclusion

Solving quadratic equations can be a challenging task, but with the quadratic formula, we can find the solutions with ease. In this article, we have solved the equation $x^2 - 4x - 32 = 0$ using the quadratic formula and verified the solutions by plugging them back into the original equation. We hope that this article has provided a clear and concise guide to solving quadratic equations.

Frequently Asked Questions

• 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.
• How do I use the quadratic formula? To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the given equation and plug them into the formula.
• What are the coefficients in a quadratic equation? The coefficients in a quadratic equation are the numbers in front of the variable and the constant term.
• How do I check the solutions? To check the solutions, you can plug them back into the original equation to verify that they satisfy the equation.

Additional Resources

• Quadratic Formula Calculator: A online calculator that can help you solve quadratic equations using the quadratic formula.
• Quadratic Equation Solver: A online tool that can help you solve quadratic equations and provide step-by-step solutions.
• Mathematics Textbook: A comprehensive textbook that covers quadratic equations and other mathematical concepts.

References

• Quadratic Formula: A Wikipedia article that provides a detailed explanation of the quadratic formula and its applications.
• Quadratic Equation: A Math Open Reference article that provides a clear and concise explanation of quadratic equations and their solutions.
• Mathematics: A Khan Academy course that covers quadratic equations and other mathematical concepts.
  

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed how to solve quadratic equations using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations and provide additional resources for further learning.

Q&A

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 use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the given equation and plug them into the formula. The quadratic formula is given by:

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

Q: What are the coefficients in a quadratic equation?

A: The coefficients in a quadratic equation are the numbers in front of the variable and the constant term. In the equation $x^2 - 4x - 32 = 0$, the coefficients are $a = 1$, $b = -4$, and $c = -32$.

Q: How do I check the solutions?

A: To check the solutions, you can plug them back into the original equation to verify that they satisfy the equation. For example, if we have the solution $x = 8$, we can plug it back into the equation $x^2 - 4x - 32 = 0$ to verify that it satisfies the equation.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable is one. A quadratic equation, on the other hand, is a polynomial equation of degree two, which means the highest power of the variable is two.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring if it can be factored into the product of two binomials. For example, the equation $x^2 - 4x - 32 = 0$ can be factored as $(x - 8)(x + 4) = 0$.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$. The discriminant determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Additional Resources

• Quadratic Formula Calculator: A online calculator that can help you solve quadratic equations using the quadratic formula.
• Quadratic Equation Solver: A online tool that can help you solve quadratic equations and provide step-by-step solutions.
• Mathematics Textbook: A comprehensive textbook that covers quadratic equations and other mathematical concepts.

References

• Quadratic Formula: A Wikipedia article that provides a detailed explanation of the quadratic formula and its applications.
• Quadratic Equation: A Math Open Reference article that provides a clear and concise explanation of quadratic equations and their solutions.
• Mathematics: A Khan Academy course that covers quadratic equations and other mathematical concepts.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we have answered some frequently asked questions about quadratic equations and provided additional resources for further learning. We hope that this article has provided a clear and concise guide to quadratic equations and their applications.