What Are The Solution(s) Of The Quadratic Equation $98-x^2=0$?A. $x= \pm 2 \sqrt{7}$B. $x= \pm 6 \sqrt{3}$C. $x= \pm 7 \sqrt{2}$D. No Real Solution

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 $98-x^2=0$. This equation is a classic example of a quadratic equation in the form of $ax^2+bx+c=0$, where $a$, $b$, and $c$ are constants. Our goal is to find the solutions to this equation, which will give us the values of $x$ that satisfy the equation.

Understanding the Quadratic Formula

Before we dive into solving the equation, let's review the quadratic formula. The quadratic formula is a powerful tool that allows us to find the solutions to a quadratic equation in the form of $ax^2+bx+c=0$. The formula is given by:

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

In our case, the equation is $98-x^2=0$, which can be rewritten as $x^2-98=0$. Comparing this equation to the standard form, we have $a=1$, $b=0$, and $c=-98$.

Applying the Quadratic Formula

Now that we have identified the values of $a$, $b$, and $c$, we can plug them into the quadratic formula. We get:

$x = \frac{-0 \pm \sqrt{0^2-4(1)(-98)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{\pm \sqrt{392}}{2}$

Simplifying the Square Root

The square root of $392$ can be simplified by factoring it as $392 = 4 \times 98 = 4 \times 2 \times 49 = 4 \times 2 \times 7^2$. Therefore, we can write:

$\sqrt{392} = \sqrt{4 \times 2 \times 7^2} = 2 \times 7 \sqrt{2}$

Finding the Solutions

Now that we have simplified the square root, we can substitute it back into the expression for $x$. We get:

$x = \frac{\pm 2 \times 7 \sqrt{2}}{2}$

Simplifying the expression, we get:

$x = \pm 7 \sqrt{2}$

Conclusion

In this article, we have solved the quadratic equation $98-x^2=0$ using the quadratic formula. We have identified the values of $a$, $b$, and $c$, plugged them into the formula, and simplified the expression to find the solutions. The solutions to the equation are $x = \pm 7 \sqrt{2}$.

Answer

The correct answer is:

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Introduction

In our previous article, we solved the quadratic equation $98-x^2=0$ using the quadratic formula. In this article, we will answer some frequently asked questions related to quadratic equations. Whether you are a student, teacher, or professional, this article will provide you with a comprehensive understanding of quadratic equations and their solutions.

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 (usually $x$) is two. The general form of a quadratic equation is $ax^2+bx+c=0$, where $a$, $b$, and $c$ are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool that allows us to find the solutions to a quadratic equation in the form of $ax^2+bx+c=0$. The formula is given by:

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

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, plug these values into the formula and simplify the expression to find the solutions.

Q: What is the difference between real and complex solutions?

A: Real solutions are values of $x$ that satisfy the equation and are real numbers. Complex solutions, on the other hand, are values of $x$ that satisfy the equation but are complex numbers, which means they have an imaginary part.

Q: How do I determine if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to examine the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2-4ac$. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: How do I simplify the square root in the quadratic formula?

A: To simplify the square root in the quadratic formula, you need to factor the expression under the square root into perfect squares. Then, take the square root of each perfect square and simplify the expression.

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

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

• Not identifying the values of $a$, $b$, and $c$ correctly
• Not simplifying the expression under the square root correctly
• Not checking if the solutions are real or complex
• Not using the correct formula or method to solve the equation

Conclusion

In this article, we have answered some frequently asked questions related to quadratic equations. Whether you are a student, teacher, or professional, this article will provide you with a comprehensive understanding of quadratic equations and their solutions. Remember to always identify the values of $a$, $b$, and $c$ correctly, simplify the expression under the square root correctly, and check if the solutions are real or complex.

Additional Resources

For more information on quadratic equations, we recommend the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Answer

The correct answers to the questions are:

1. A quadratic equation is a polynomial equation of degree two.
2. The quadratic formula is a powerful tool that allows us to find the solutions to a quadratic equation in the form of $ax^2+bx+c=0$.
3. To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation and plug them into the formula.
4. Real solutions are values of $x$ that satisfy the equation and are real numbers.
5. The discriminant is the expression under the square root in the quadratic formula, which is $b^2-4ac$.
6. To simplify the square root in the quadratic formula, you need to factor the expression under the square root into perfect squares.
7. Some common mistakes to avoid when solving quadratic equations include not identifying the values of $a$, $b$, and $c$ correctly, not simplifying the expression under the square root correctly, not checking if the solutions are real or complex, and not using the correct formula or method to solve the equation.