Consider The Following Equation Of A Quadratic: F ( X ) = 1 ( X − 3 ) 2 + 3 F(x) = 1(x-3)^2 + 3 F ( X ) = 1 ( X − 3 ) 2 + 3 Write The Equation In General Form: Y = □ X 2 + □ X + □ Y = \square X^2 + \square X + \square Y = □ X 2 + □ X + □

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving quadratic equations, specifically the equation f(x)=1(x−3)2+3f(x) = 1(x-3)^2 + 3. We will rewrite the equation in general form and explore the various methods for solving quadratic equations.

Understanding the Given Equation

The given equation is f(x)=1(x−3)2+3f(x) = 1(x-3)^2 + 3. This equation represents a quadratic function, which is a polynomial of degree two. The general form of a quadratic equation is y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants.

Expanding the Given Equation

To rewrite the equation in general form, we need to expand the squared term. Using the formula (x−3)2=x2−6x+9(x-3)^2 = x^2 - 6x + 9, we can rewrite the equation as:

f(x)=1(x2−6x+9)+3f(x) = 1(x^2 - 6x + 9) + 3

Distributing the Coefficient

Now, we can distribute the coefficient 11 to the terms inside the parentheses:

f(x)=x2−6x+9+3f(x) = x^2 - 6x + 9 + 3

Combining Like Terms

Finally, we can combine the like terms to simplify the equation:

f(x)=x2−6x+12f(x) = x^2 - 6x + 12

Rewriting the Equation in General Form

Now that we have expanded and simplified the equation, we can rewrite it in general form:

y=x2−6x+12y = x^2 - 6x + 12

In this form, we can easily identify the coefficients aa, bb, and cc. In this case, a=1a = 1, b=−6b = -6, and c=12c = 12.

Methods for Solving Quadratic Equations

There are several methods for solving quadratic equations, including:

Factoring

Factoring is a method of solving quadratic equations by expressing the equation as a product of two binomials. For example, the equation y=x2−6x+12y = x^2 - 6x + 12 can be factored as:

y=(x−3)(x−4)y = (x - 3)(x - 4)

Quadratic Formula

The quadratic formula is a method of solving quadratic equations by using the formula:

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

In this case, a=1a = 1, b=−6b = -6, and c=12c = 12. Plugging these values into the formula, we get:

x=−(−6)±(−6)2−4(1)(12)2(1)x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(12)}}{2(1)}

x=6±36−482x = \frac{6 \pm \sqrt{36 - 48}}{2}

x=6±−122x = \frac{6 \pm \sqrt{-12}}{2}

Graphing

Graphing is a method of solving quadratic equations by plotting the graph of the equation. The graph of the equation y=x2−6x+12y = x^2 - 6x + 12 is a parabola that opens upward.

Conclusion

In this article, we have explored the concept of quadratic equations and solved the equation f(x)=1(x−3)2+3f(x) = 1(x-3)^2 + 3 in general form. We have also discussed various methods for solving quadratic equations, including factoring, the quadratic formula, and graphing. By understanding these methods, we can solve quadratic equations and apply them to real-world problems.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields, including:

Physics

Quadratic equations are used to model the motion of objects under the influence of gravity. For example, the equation y=x2−6x+12y = x^2 - 6x + 12 can be used to model the trajectory of a projectile.

Engineering

Quadratic equations are used to design and optimize systems, such as bridges and buildings. For example, the equation y=x2−6x+12y = x^2 - 6x + 12 can be used to design a bridge that spans a certain distance.

Economics

Quadratic equations are used to model economic systems, such as supply and demand curves. For example, the equation y=x2−6x+12y = x^2 - 6x + 12 can be used to model the demand for a certain product.

Real-World Examples

Quadratic equations have numerous real-world applications, including:

Projectile Motion

The equation y=x2−6x+12y = x^2 - 6x + 12 can be used to model the trajectory of a projectile, such as a baseball or a basketball.

Bridge Design

The equation y=x2−6x+12y = x^2 - 6x + 12 can be used to design a bridge that spans a certain distance.

Supply and Demand

The equation y=x2−6x+12y = x^2 - 6x + 12 can be used to model the demand for a certain product, such as a commodity or a service.

Conclusion

Frequently Asked Questions

Quadratic equations can be a challenging topic, but with the right guidance, you can master them. Here are some frequently asked questions about quadratic equations, along with their answers.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two. The general form of a quadratic equation is y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including factoring, the quadratic formula, and graphing. The method you choose will depend on the specific equation and the information you need to find.

Q: What is the quadratic formula?

A: The quadratic formula is a method of solving quadratic equations by using the formula:

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, but it requires that you know the values of aa, bb, and cc.

Q: How do I factor a quadratic equation?

A: Factoring a quadratic equation involves expressing it as a product of two binomials. For example, the equation y=x2−6x+12y = x^2 - 6x + 12 can be factored as:

y=(x−3)(x−4)y = (x - 3)(x - 4)

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. The general form of a linear equation is y=mx+by = mx + b, where mm and bb are constants.

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 formula function that you can use to find the solutions to a quadratic equation.

Q: How do I graph a quadratic equation?

A: Graphing a quadratic equation involves plotting the graph of the equation on a coordinate plane. The graph of a quadratic equation is a parabola that opens upward or downward.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including modeling the motion of objects, designing systems, and analyzing economic systems. Some examples of real-world applications of quadratic equations include:

  • Modeling the trajectory of a projectile
  • Designing a bridge that spans a certain distance
  • Analyzing the demand for a certain product

Q: Can I use quadratic equations to solve problems in physics?

A: Yes, you can use quadratic equations to solve problems in physics. Quadratic equations are used to model the motion of objects under the influence of gravity, and they can be used to find the solutions to problems involving projectile motion.

Q: Can I use quadratic equations to solve problems in engineering?

A: Yes, you can use quadratic equations to solve problems in engineering. Quadratic equations are used to design and optimize systems, and they can be used to find the solutions to problems involving bridge design and other engineering applications.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the methods for solving quadratic equations, you can apply them to real-world problems and make informed decisions. Whether it's modeling the motion of objects, designing systems, or analyzing economic systems, quadratic equations play a crucial role in many areas of life.