Solve X 2 = 63 X^2=63 X 2 = 63 , Where X X X Is A Real Number. Simplify Your Answer As Much As Possible.If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, Click No Solution. X = X = X =

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 $x^2=63$, where $x$ is a real number. We will break down the solution process into manageable steps and provide a clear explanation of each step.

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. In our case, the equation is $x^2=63$, which can be rewritten as $x^2-63=0$.

Step 1: Factor the Equation

To solve the equation $x^2-63=0$, we need to factor the left-hand side. Factoring involves expressing the quadratic expression as a product of two binomials. In this case, we can factor the equation as $(x-√63)(x+√63)=0$.

Step 2: Apply the Zero Product Property

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. In this case, we have $(x-√63)(x+√63)=0$, which means that either $(x-√63)=0$ or $(x+√63)=0$.

Step 3: Solve for $x$

Now that we have two separate equations, we can solve for $x$ in each case. For the first equation, $(x-√63)=0$, we can add $√63$ to both sides to get $x=√63$. For the second equation, $(x+√63)=0$, we can subtract $√63$ from both sides to get $x=-√63$.

Step 4: Simplify the Solutions

We have found two solutions for $x$: $x=√63$ and $x=-√63$. However, we can simplify these solutions further by rationalizing the denominator. To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of $√63$ is $-√63$. Multiplying both the numerator and denominator by $-√63$, we get:

$x=√63 = √63 \times \frac{-√63}{-√63} = \frac{-63}{-√63} = \frac{63}{√63}$

$x=-√63 = -√63 \times \frac{-√63}{-√63} = \frac{63}{√63}$

Step 5: Write the Final Answer

We have found two solutions for $x$: $x=√63$ and $x=-√63$. However, we can simplify these solutions further by rationalizing the denominator. Multiplying both the numerator and denominator by $√63$, we get:

$x=√63 = \frac{63}{√63} \times \frac{√63}{√63} = \frac{63√63}{63} = √63$

$x=-√63 = \frac{63}{√63} \times \frac{√63}{√63} = \frac{63√63}{63} = -√63$

Therefore, the final answer is:

$x = √63, -√63$

Conclusion

Solving quadratic equations can be a challenging task, but with the right approach and techniques, it can be made easier. In this article, we have solved the quadratic equation $x^2=63$ and found two real solutions: $x=√63$ and $x=-√63$. We have also simplified these solutions further by rationalizing the denominator. With practice and patience, you can master the art of solving quadratic equations and tackle more complex problems with confidence.

Frequently Asked Questions

• 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.
• Q: How do I factor a quadratic equation? A: To factor a quadratic equation, you need to express the quadratic expression as a product of two binomials.
• Q: What is the zero product property? A: The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero.
• Q: How do I rationalize the denominator? A: To rationalize the denominator, you multiply both the numerator and denominator by the conjugate of the denominator.

Additional Resources

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

Note: The above article is a rewritten version of the original content, optimized for readability and SEO. The article includes headings, subheadings, and bullet points to make it easier to read and understand. The article also includes a conclusion, frequently asked questions, and additional resources to provide more value to the reader.

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task. In our previous article, we provided a step-by-step guide to solving the quadratic equation $x^2=63$. However, we understand that you may still have questions about quadratic equations. In this article, we will address some of the most frequently asked questions about quadratic equations and provide clear and concise answers.

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.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to express the quadratic expression as a product of two binomials. This can be done by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the zero product property?

A: The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This means that if you have a quadratic equation in the form of $(x-a)(x-b)=0$, then either $(x-a)=0$ or $(x-b)=0$.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Apply the zero product property to find the solutions.
3. Simplify the solutions, if necessary.

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 (in this case, $x$) is one. A quadratic equation, on the other hand, is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic equation solver that can help you find the solutions.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is the expression under the square root in the quadratic formula. It is calculated as $b^2-4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What does the discriminant tell me about the solutions of a quadratic equation?

A: The discriminant tells you whether the quadratic equation has real or complex solutions. If the discriminant is positive, then the quadratic equation has two real solutions. If the discriminant is zero, then the quadratic equation has one real solution. If the discriminant is negative, then the quadratic equation has no real solutions.

Q: Can I use the quadratic formula to solve a quadratic equation?

A: Yes, you can use the quadratic formula to solve a quadratic equation. The quadratic formula is given by:

$x=\frac{-b±\sqrt{b^2-4ac}}{2a}$

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve a quadratic equation. It is given by:

$x=\frac{-b±\sqrt{b^2-4ac}}{2a}$

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. The quadratic formula will give you the complex solutions in the form of $x=a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: What is the imaginary unit?

A: The imaginary unit is a mathematical concept that is used to represent the square root of -1. It is denoted by the letter $i$ and is defined as $i^2=-1$.

Q: Can I use the quadratic formula to solve a quadratic equation with rational solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with rational solutions. The quadratic formula will give you the rational solutions in the form of $x=\frac{p}{q}$, where $p$ and $q$ are integers.

Q: What is the difference between a rational solution and an irrational solution?

A: A rational solution is a solution that can be expressed as a ratio of integers, while an irrational solution is a solution that cannot be expressed as a ratio of integers.

Q: Can I use the quadratic formula to solve a quadratic equation with repeated solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with repeated solutions. The quadratic formula will give you the repeated solutions in the form of $x=a$, where $a$ is a real number.

Q: What is the difference between a repeated solution and a distinct solution?

A: A repeated solution is a solution that is repeated, while a distinct solution is a solution that is unique.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions and repeated solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions and repeated solutions. The quadratic formula will give you the complex solutions and repeated solutions in the form of $x=a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Conclusion

Quadratic equations can be a challenging topic, but with the right approach and techniques, you can master the art of solving them. In this article, we have addressed some of the most frequently asked questions about quadratic equations and provided clear and concise answers. We hope that this article has been helpful in clarifying any doubts you may have had about quadratic equations.

Additional Resources

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

Note: The above article is a rewritten version of the original content, optimized for readability and SEO. The article includes headings, subheadings, and bullet points to make it easier to read and understand. The article also includes a conclusion and additional resources to provide more value to the reader.