Multiplying Out the Expression

Oct 13, 2025 by windailys.com 114 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. 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 finding the equation that has a solution at $2$ and $2$ . This involves multiplying out the given expression and simplifying it to obtain the desired quadratic equation.

Understanding the Problem

The problem requires us to find the equation that has a solution at $2$ and $2$ . This means that when we substitute $x=2$ into the equation, we should get a true statement. In other words, the equation should be satisfied when $x=2$ . To achieve this, we need to multiply out the given expression and simplify it to obtain the desired quadratic equation.

Multiplying Out the Expression

To multiply out the expression $(x-2)(x-2)$ , we need to use the distributive property of multiplication over addition. This means that we need to multiply each term in the first expression by each term in the second expression and then combine like terms.

(x-2)(x-2) = x^2 - 2x - 2x + 4

Simplifying the Expression

Now that we have multiplied out the expression, we need to simplify it to obtain the desired quadratic equation. To do this, we need to combine like terms. In this case, we have two terms with the variable $x$ , which are $-2x$ and $-2x$ . We can combine these terms by adding their coefficients.

x^2 - 2x - 2x + 4 = x^2 - 4x + 4

The Final Equation

The final equation is $x^2 - 4x + 4$ . This is the equation that has a solution at $2$ and $2$ . When we substitute $x=2$ into this equation, we get:

(2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0

This shows that the equation $x^2 - 4x + 4$ is satisfied when $x=2$ , which means that $2$ is a solution to this equation.

Conclusion

In this article, we have found the equation that has a solution at $2$ and $2$ . We started by multiplying out the given expression and then simplified it to obtain the desired quadratic equation. The final equation is $x^2 - 4x + 4$ , which is satisfied when $x=2$ . This demonstrates the importance of quadratic equations in mathematics and their numerous applications in various fields.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of quadratic equations include:

Projectile Motion: Quadratic equations are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
Signal Processing: Quadratic equations are used in signal processing to filter signals and remove noise.
Computer Graphics: Quadratic equations are used in computer graphics to create 3D models and animations.

Solving Quadratic Equations

Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing. The quadratic formula is a popular method for solving quadratic equations, and it is given by:

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

This formula can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Graphing Quadratic Equations

Quadratic equations can be graphed using various methods, including plotting points, using a graphing calculator, or using a computer program. The graph of a quadratic equation is a parabola, which is a U-shaped curve. The vertex of the parabola is the point where the curve changes direction, and it is given by:

x = -\frac{b}{2a}

This formula can be used to find the x-coordinate of the vertex of the parabola.

Conclusion

In this article, we have discussed the equation that has a solution at $2$ and $2$ . We started by multiplying out the given expression and then simplified it to obtain the desired quadratic equation. The final equation is $x^2 - 4x + 4$ , which is satisfied when $x=2$ . We have also discussed the applications of quadratic equations and the methods for solving and graphing them. Quadratic equations are an important concept in mathematics, and they have numerous applications in various fields.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In our previous article, we discussed the equation that has a solution at $2$ and $2$ . In this article, we will provide a Q&A guide to help you understand quadratic equations better.

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 $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The quadratic formula is a popular method for solving quadratic equations, and it is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . 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 plug in the values of $a$ , $b$ , and $c$ into the formula. Then, you need to simplify the expression and solve for $x$ .

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the curve changes direction. It is given by:

x = -\frac{b}{2a}

Q: How do I find the x-coordinate of the vertex of a parabola?

A: To find the x-coordinate of the vertex of a parabola, you need to plug in the values of $a$ and $b$ into the formula:

x = -\frac{b}{2a}

Q: What is the y-coordinate of the vertex of a parabola?

A: The y-coordinate of the vertex of a parabola is given by:

y = f(-\frac{b}{2a})

Q: How do I graph a quadratic equation?

A: There are several methods to graph a quadratic equation, including plotting points, using a graphing calculator, or using a computer program. The graph of a quadratic equation is a parabola, which is a U-shaped curve.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of quadratic equations include:

Projectile Motion: Quadratic equations are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
Signal Processing: Quadratic equations are used in signal processing to filter signals and remove noise.
Computer Graphics: Quadratic equations are used in computer graphics to create 3D models and animations.

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 need to examine 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.

Conclusion

In this article, we have provided a Q&A guide to help you understand quadratic equations better. We have discussed the definition of a quadratic equation, the methods for solving quadratic equations, and the applications of quadratic equations. We have also provided answers to frequently asked questions about quadratic equations.