Solve $x = 5x^2$.a. $x = 0$b. $x = 0$ Or $\frac{1}{5}$c. $x = \frac{1}{5}$d. $x = 5$

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 $x = 5x^2$. This equation may seem simple, but it requires a careful approach to arrive at the correct solution. We will break down the solution step by step, using algebraic manipulations and mathematical reasoning.

Understanding the Equation

The given equation is $x = 5x^2$. This is a quadratic equation in the variable $x$, where the coefficient of $x^2$ is $5$. To solve this equation, we need to isolate the variable $x$.

Step 1: Rearranging the Equation

The first step in solving the equation is to rearrange it to get all the terms on one side of the equation. We can do this by subtracting $5x^2$ from both sides of the equation.

$$x - 5x^2 = 0$$

Step 2: Factoring the Equation

The next step is to factor the equation. We can factor out the common term $x$ from the left-hand side of the equation.

$$x(1 - 5x) = 0$$

Step 3: Applying the Zero Product Property

The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. We can apply this property to the factored equation.

$$x(1 - 5x) = 0$$

This implies that either $x = 0$ or $1 - 5x = 0$.

Step 4: Solving for $x$

We can solve for $x$ by setting each factor equal to zero.

$$x = 0$$

or

$$1 - 5x = 0$$

Solving for $x$ in the second equation, we get:

$$1 - 5x = 0$$

$$-5x = -1$$

$$x = \frac{1}{5}$$

Conclusion

In conclusion, the solutions to the quadratic equation $x = 5x^2$ are $x = 0$ and $x = \frac{1}{5}$. These solutions can be obtained by applying the zero product property and solving for $x$.

Final Answer

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Introduction

In our previous article, we solved the quadratic equation $x = 5x^2$ and arrived at the solutions $x = 0$ and $x = \frac{1}{5}$. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use various methods such as factoring, the quadratic formula, or graphing. The method you choose will depend on the specific equation and the type of solution you are looking for.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

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 quadratic equation. Then, you can plug these values into the formula and simplify to find the solutions.

Q: What is the difference between the two solutions of a quadratic equation?

A: The two solutions of a quadratic equation are called the roots or the zeros of the equation. They are the values of the variable that make the equation true. In some cases, the solutions may be real and distinct, while in other cases, they may be complex or repeated.

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

A: No, a quadratic equation can have at most two solutions. This is because the graph of a quadratic equation is a parabola, which has a maximum or minimum point, but no more than two intersections with the x-axis.

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

A: To determine the number of solutions of a quadratic equation, you can use the discriminant, which is given by $b^2 - 4ac$. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: Can a quadratic equation have no real solutions?

A: Yes, a quadratic equation can have no real solutions. This occurs when the discriminant is negative, which means that the equation has complex solutions.

Conclusion

In conclusion, solving quadratic equations requires a good understanding of the concepts and techniques involved. By using the quadratic formula, factoring, or graphing, you can find the solutions to a quadratic equation. Remember to always check the number of solutions and the nature of the solutions to ensure that you have found the correct answer.

Final Answer

The final answer is: $\boxed{b}$