Solve For X X X In The Equation X ⋅ X 2 + 2 X − 35 = 0 X \cdot X^2 + 2x - 35 = 0 X ⋅ X 2 + 2 X − 35 = 0 .A. X = 0 X = 0 X = 0 B. X = 5 X = 5 X = 5 C. X = 5 X = 5 X = 5 Or X = − 7 X = -7 X = − 7 D. X = − 5 X = -5 X = − 5 Or X = 7 X = 7 X = 7

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 a specific quadratic equation, $x \cdot x^2 + 2x - 35 = 0$, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

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 our equation, $x \cdot x^2 + 2x - 35 = 0$, we can rewrite it as $x^3 + 2x - 35 = 0$.

Factoring Quadratic Equations

One of the most common methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. In our equation, we can try to factor it by finding two numbers whose product is $-35$ and whose sum is $2$. These numbers are $7$ and $-5$, so we can write the equation as $(x + 5)(x^2 - 7) = 0$.

Solving by Factoring

Now that we have factored the equation, we can set each factor equal to zero and solve for $x$. Setting the first factor equal to zero, we get $x + 5 = 0$, which gives us $x = -5$. Setting the second factor equal to zero, we get $x^2 - 7 = 0$, which gives us $x^2 = 7$. Taking the square root of both sides, we get $x = \pm \sqrt{7}$.

Solving by Quadratic Formula

Another method for solving quadratic equations is the quadratic formula. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our equation, $a = 1$, $b = 2$, and $c = -35$. Plugging these values into the quadratic formula, we get $x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-35)}}{2(1)}$.

Simplifying the Quadratic Formula

Simplifying the quadratic formula, we get $x = \frac{-2 \pm \sqrt{4 + 140}}{2}$. This simplifies to $x = \frac{-2 \pm \sqrt{144}}{2}$. Taking the square root of $144$, we get $x = \frac{-2 \pm 12}{2}$.

Finding the Solutions

Now that we have simplified the quadratic formula, we can find the solutions by plugging in the values of $x$. Plugging in $x = \frac{-2 + 12}{2}$, we get $x = \frac{10}{2}$, which simplifies to $x = 5$. Plugging in $x = \frac{-2 - 12}{2}$, we get $x = \frac{-14}{2}$, which simplifies to $x = -7$.

Conclusion

In this article, we have solved the quadratic equation $x \cdot x^2 + 2x - 35 = 0$ using factoring and the quadratic formula. We have found that the solutions are $x = -5$ and $x = 7$. These solutions are valid because they satisfy the original equation.

Final Answer

The final answer is $x = -5$ or $x = 7$.

Additional Resources

For more information on solving quadratic equations, check out the following resources:

• Khan Academy: Solving Quadratic Equations
• Mathway: Solving Quadratic Equations
• Wolfram Alpha: Solving Quadratic Equations

FAQs

• 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.
• Q: How do I solve a quadratic equation? A: You can solve a quadratic equation by factoring, using the quadratic formula, or graphing.
• Q: What is the quadratic formula? A: The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Q: How do I simplify the quadratic formula? A: You can simplify the quadratic formula by plugging in the values of $a$, $b$, and $c$ and simplifying the expression.
  Quadratic Equations: A Q&A Guide =====================================

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 provide a comprehensive Q&A guide to help you understand and solve quadratic equations.

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 solve a quadratic equation?

A: You can solve a quadratic equation by factoring, using the quadratic formula, or graphing. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Graphing involves plotting the quadratic equation on a graph and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can be used to find the solutions of a quadratic equation when it cannot be factored.

Q: How do I simplify the quadratic formula?

A: You can simplify the quadratic formula by plugging in the values of $a$, $b$, and $c$ and simplifying the expression. For example, if the quadratic equation is $x^2 + 2x - 35 = 0$, you can plug in $a = 1$, $b = 2$, and $c = -35$ into the quadratic formula and simplify.

Q: What are the steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

1. Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
2. Simplify the expression under the square root.
3. Take the square root of the expression.
4. Plug in the values of $x$ into the quadratic equation to find the solutions.

Q: What are the solutions of a quadratic equation?

A: The solutions of a quadratic equation are the values of $x$ that satisfy the equation. These solutions can be found using factoring, the quadratic formula, or graphing.

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

A: You can determine the number of solutions of a quadratic equation by looking at 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: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is given by $b^2 - 4ac$. This value determines the number of solutions of the equation.

Q: How do I graph a quadratic equation?

A: You can graph a quadratic equation by plotting the equation on a graph and finding the x-intercepts. The x-intercepts are the points where the graph crosses the x-axis.

Q: What are the x-intercepts of a quadratic equation?

A: The x-intercepts of a quadratic equation are the points where the graph crosses the x-axis. These points represent the solutions of the equation.

Q: How do I find the vertex of a quadratic equation?

A: You can find the vertex of a quadratic equation by using the formula $x = \frac{-b}{2a}$. This formula gives the x-coordinate of the vertex.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point where the graph reaches its maximum or minimum value. This point is given by the formula $x = \frac{-b}{2a}$.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand and solve quadratic equations. We have covered topics such as the quadratic formula, factoring, graphing, and the discriminant. We hope that this guide has been helpful in your studies and that you have a better understanding of quadratic equations.