Check All Solutions To The Equation. If There Are No Solutions, Check None. X 2 = 16 X^2 = 16 X 2 = 16 A. 1 B. -4 C. 0 D. 16 E. 4 F. None

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Introduction

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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 explore the process of solving quadratic equations, with a focus on the equation $x^2 = 16$. We will break down the solution into manageable steps, and provide a clear explanation of each step.

Understanding Quadratic Equations

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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. In the equation $x^2 = 16$, we can rewrite it as:

$$x^2 - 16 = 0$$

This is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = 0$, and $c = -16$.

Solving the Equation

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To solve the equation $x^2 = 16$, we need to find the values of $x$ that satisfy the equation. We can start by factoring the left-hand side of the equation:

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

This tells us that either $(x - 4) = 0$ or $(x + 4) = 0$. We can solve each of these equations separately:

Solving $(x - 4) = 0$

To solve the equation $(x - 4) = 0$, we need to isolate the variable $x$. We can do this by adding 4 to both sides of the equation:

$$x - 4 + 4 = 0 + 4$$

This simplifies to:

$$x = 4$$

Solving $(x + 4) = 0$

To solve the equation $(x + 4) = 0$, we need to isolate the variable $x$. We can do this by subtracting 4 from both sides of the equation:

$$x + 4 - 4 = 0 - 4$$

This simplifies to:

$$x = -4$$

Checking the Solutions

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Now that we have found the solutions to the equation, we need to check if they are correct. We can do this by plugging the solutions back into the original equation:

Checking $x = 4$

We can plug $x = 4$ back into the original equation:

$$4^2 = 16$$

This is true, so $x = 4$ is a solution to the equation.

Checking $x = -4$

We can plug $x = -4$ back into the original equation:

$$(-4)^2 = 16$$

This is also true, so $x = -4$ is a solution to the equation.

Conclusion

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In this article, we have solved the quadratic equation $x^2 = 16$ using factoring. We found that the solutions to the equation are $x = 4$ and $x = -4$. We also checked these solutions by plugging them back into the original equation. This confirms that the solutions are correct.

Final Answer

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The final answer to the equation $x^2 = 16$ is:

• A. 1: Incorrect
• B. -4: Correct
• C. 0: Incorrect
• D. 16: Incorrect
• E. 4: Correct
• F. None: Incorrect

The correct answers are B and E.

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Introduction

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Quadratic equations can be a challenging topic for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations, including their definition, solving methods, and applications.

Q: What is a quadratic equation?

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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?

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A: There are several methods to solve a quadratic equation, including:

• Factoring: This involves expressing the quadratic 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 graphing the quadratic equation on a coordinate plane and finding the x-intercepts.

Q: What is the quadratic formula?

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

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A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. You also need to simplify the expression under the square root.

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

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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 variable, while a linear equation does not.

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

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A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula always produces two solutions, and there is no way to have more than two solutions.

Q: Can a quadratic equation have no solutions?

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A: Yes, a quadratic equation can have no solutions. This occurs when the expression under the square root in the quadratic formula is negative.

Q: What is the significance of the discriminant in a quadratic equation?

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A: The discriminant is the expression under the square root in the quadratic formula. It 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.

Q: Can a quadratic equation be used to model real-world problems?

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A: Yes, quadratic equations can be used to model a wide range of real-world problems, including projectile motion, optimization problems, and electrical circuits.

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

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

Conclusion

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In this article, we have addressed some of the most frequently asked questions about quadratic equations. We have covered topics such as the definition of a quadratic equation, solving methods, and applications. We hope that this article has provided you with a better understanding of quadratic equations and their significance in mathematics and real-world problems.

Final Answer

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The final answer to the question "What is a quadratic equation?" is:

• 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.

The final answer to the question "How do I solve a quadratic equation?" is:

• There are several methods to solve a quadratic equation, including factoring, quadratic formula, and graphing.

The final answer to the question "What is the quadratic formula?" is:

• The quadratic formula is a formula used to find the solutions to a quadratic equation. 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.