Solve The Equation:$ (x+2)(x-1) = -x^2 + 1 $

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Introduction

In mathematics, solving quadratic equations is a fundamental concept that helps us find the values of unknown variables. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on solving the quadratic equation (x+2)(x−1)=−x2+1(x+2)(x-1) = -x^2 + 1. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Equation

The given equation is (x+2)(x−1)=−x2+1(x+2)(x-1) = -x^2 + 1. To solve this equation, we need to simplify it and isolate the variable xx. Let's start by expanding the left-hand side of the equation using the distributive property.

Expanding the Left-Hand Side

import sympy as sp

x = sp.symbols('x')



left_hand_side = sp.expand((x+2)*(x-1))


print(left_hand_side)

The output of the above code is x2+x−2x^2 + x - 2. Now, let's rewrite the original equation with the expanded left-hand side.

(x2+x−2)=−x2+1(x^2 + x - 2) = -x^2 + 1

Simplifying the Equation

To simplify the equation, we can combine like terms. Let's move all the terms to the left-hand side of the equation.

x2+x−2+x2−1=0x^2 + x - 2 + x^2 - 1 = 0

Combine like terms:

2x2+x−3=02x^2 + x - 3 = 0

Solving the Quadratic Equation

Now that we have simplified the equation, we can use various methods to solve for xx. Let's use the quadratic formula, which is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a=2a = 2, b=1b = 1, and c=−3c = -3. Plugging these values into the quadratic formula, we get:

x=−1±12−4(2)(−3)2(2)x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-3)}}{2(2)}

x=−1±1+244x = \frac{-1 \pm \sqrt{1 + 24}}{4}

x=−1±254x = \frac{-1 \pm \sqrt{25}}{4}

x=−1±54x = \frac{-1 \pm 5}{4}

Finding the Solutions

Now, let's find the two possible solutions for xx.

x1=−1+54=44=1x_1 = \frac{-1 + 5}{4} = \frac{4}{4} = 1

x2=−1−54=−64=−32x_2 = \frac{-1 - 5}{4} = \frac{-6}{4} = -\frac{3}{2}

Conclusion

In this article, we solved the quadratic equation (x+2)(x−1)=−x2+1(x+2)(x-1) = -x^2 + 1 using the quadratic formula. We simplified the equation, combined like terms, and used the quadratic formula to find the two possible solutions for xx. The solutions are x=1x = 1 and x=−32x = -\frac{3}{2}. We hope this article has helped you understand how to solve quadratic equations and has provided you with a step-by-step guide to solving the given equation.

Additional Resources

Frequently Asked Questions

  • What is a quadratic equation? A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two.
  • How do I solve a quadratic equation? You can use various methods to solve a quadratic equation, including the quadratic formula, factoring, and completing the square.
  • What is the quadratic formula? The quadratic formula is a formula that helps you solve quadratic equations. It is given by: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
    Quadratic Equation Q&A: Frequently Asked Questions =====================================================

Introduction

In our previous article, we solved the quadratic equation (x+2)(x−1)=−x2+1(x+2)(x-1) = -x^2 + 1 using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations. Whether you are a student, teacher, or just someone who wants to learn more about quadratic equations, this article is for you.

Q&A

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. It is typically written in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

A: You can use various methods to solve a quadratic equation, including the quadratic formula, factoring, and completing the square. The quadratic formula is a formula that helps you solve quadratic equations. It is given by: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that helps you solve quadratic equations. It is given by: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. This formula can be used to find the solutions to a quadratic equation.

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of xx that make the equation true. These values can be found using the quadratic formula.

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

A: To determine the number of solutions to a quadratic equation, you can use the discriminant, which is given by b2−4acb^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?

A: The discriminant is a value that can be used to determine the number of solutions to a quadratic equation. It is given by b2−4acb^2 - 4ac.

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

A: To use the discriminant to determine the number of solutions to a quadratic equation, you can follow these steps:

  1. Calculate the discriminant by plugging in the values of aa, bb, and cc into the formula b2−4acb^2 - 4ac.
  2. If the discriminant is positive, the equation has two distinct solutions.
  3. If the discriminant is zero, the equation has one repeated solution.
  4. If the discriminant is negative, the equation has no real solutions.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or a computer program. You can also use a table of values to plot the graph.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. It is the minimum or maximum point on the graph.

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

A: To find the vertex of a quadratic equation, you can use the formula x=−b2ax = \frac{-b}{2a}. This will give you the x-coordinate of the vertex.

Q: What is the axis of symmetry of a quadratic equation?

A: The axis of symmetry of a quadratic equation is a vertical line that passes through the vertex of the parabola. It is a line of symmetry that divides the graph into two equal parts.

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

A: To find the axis of symmetry of a quadratic equation, you can use the formula x=−b2ax = \frac{-b}{2a}. This will give you the x-coordinate of the axis of symmetry.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations. We covered topics such as the quadratic formula, solutions to quadratic equations, the discriminant, and graphing quadratic equations. We hope this article has helped you understand quadratic equations better and has provided you with a comprehensive guide to solving quadratic equations.

Additional Resources

Frequently Asked Questions

  • What is a quadratic equation? A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two.
  • How do I solve a quadratic equation? You can use various methods to solve a quadratic equation, including the quadratic formula, factoring, and completing the square.
  • What is the quadratic formula? The quadratic formula is a formula that helps you solve quadratic equations. It is given by: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  • What is the discriminant? The discriminant is a value that can be used to determine the number of solutions to a quadratic equation. It is given by b2−4acb^2 - 4ac.
  • How do I use the discriminant to determine the number of solutions to a quadratic equation? To use the discriminant to determine the number of solutions to a quadratic equation, you can follow these steps: 1. Calculate the discriminant by plugging in the values of aa, bb, and cc into the formula b2−4acb^2 - 4ac. 2. If the discriminant is positive, the equation has two distinct solutions. 3. If the discriminant is zero, the equation has one repeated solution. 4. If the discriminant is negative, the equation has no real solutions.