S5 Working With Complex Numbers Test: Performance Task 11. Solve The Equation: $6x^2 + 18 = 0$. Give Exact Solution(s), No Rounding Or Decimals.2. Solve The Equation: $2x^2 - 11x + 5 = 0$. Give Exact Solution(s), No Rounding

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will delve into the world of quadratic equations and provide a step-by-step guide on how to solve them. We will focus on two specific equations: 6x2+18=06x^2 + 18 = 0 and 2x2−11x+5=02x^2 - 11x + 5 = 0. By the end of this article, you will be equipped with the knowledge and skills to tackle even the most complex quadratic equations.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) 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. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and completing the square.

Solving the Equation 6x2+18=06x^2 + 18 = 0

To solve the equation 6x2+18=06x^2 + 18 = 0, we need to isolate the variable x. The first step is to subtract 18 from both sides of the equation:

6x^2 = -18

Next, we divide both sides of the equation by 6:

x^2 = -3

Now, we take the square root of both sides of the equation:

x = ±√(-3)

Since the square root of a negative number is an imaginary number, we can rewrite the solution as:

x = ±i√3

where i is the imaginary unit, which is defined as the square root of -1.

Solving the Equation 2x2−11x+5=02x^2 - 11x + 5 = 0

To solve the equation 2x2−11x+5=02x^2 - 11x + 5 = 0, we can use the quadratic formula. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 2, b = -11, and c = 5. Plugging these values into the quadratic formula, we get:

x = (11 ± √((-11)^2 - 4(2)(5))) / 2(2)

x = (11 ± √(121 - 40)) / 4

x = (11 ± √81) / 4

x = (11 ± 9) / 4

Now, we can simplify the solution by considering the two possible cases:

Case 1: x = (11 + 9) / 4

x = 20 / 4

x = 5

Case 2: x = (11 - 9) / 4

x = 2 / 4

x = 1/2

Therefore, the solutions to the equation 2x2−11x+5=02x^2 - 11x + 5 = 0 are x = 5 and x = 1/2.

Conclusion

In this article, we have solved two quadratic equations: 6x2+18=06x^2 + 18 = 0 and 2x2−11x+5=02x^2 - 11x + 5 = 0. We have used various methods, including factoring and the quadratic formula, to find the solutions to these equations. By following the step-by-step guide provided in this article, you should be able to tackle even the most complex quadratic equations. Remember to always check your solutions by plugging them back into the original equation to ensure that they are correct.

Tips and Tricks

  • When solving quadratic equations, always start by simplifying the equation and isolating the variable x.
  • Use the quadratic formula when the equation cannot be factored easily.
  • Check your solutions by plugging them back into the original equation to ensure that they are correct.
  • Practice, practice, practice! The more you practice solving quadratic equations, the more comfortable you will become with the process.

Common Mistakes to Avoid

  • Don't forget to check your solutions by plugging them back into the original equation.
  • Make sure to simplify the equation and isolate the variable x before applying the quadratic formula.
  • Don't confuse the quadratic formula with the formula for the area of a circle (A = πr^2).
  • Be careful when simplifying expressions involving square roots.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Computer Science: Quadratic equations are used in algorithms for solving problems, such as finding the shortest path between two points.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand curves.

Final Thoughts

Introduction

Quadratic equations can be a challenging topic for many students. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important mathematical concept.

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 (in this case, x) 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: There are several methods to solve a quadratic equation, including:

  • Factoring: If the equation can be factored into the product of two binomials, you can solve it by setting each factor equal to zero.
  • Quadratic formula: If the equation cannot be factored easily, you can use the quadratic formula to find the solutions.
  • Completing the square: This method involves rewriting the equation in a form that allows you to easily find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows you to find the solutions to a quadratic equation. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

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, simplify the expression and solve for x.

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

A: A linear equation is an equation in which the highest power of the variable (in this case, x) is one. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is two.

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 quadratic formula always produces two solutions, and there is no way to have more than two solutions to a quadratic equation.

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:

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?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is given by:

b^2 - 4ac

The discriminant can be used to determine the number of solutions to a quadratic equation.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. This occurs when the discriminant is negative, and the solutions are given by:

x = (-b ± √(-4ac)) / 2a

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the following steps:

  • Find the x-intercepts of the equation by setting y = 0 and solving for x.
  • Find the vertex of the parabola by using the formula:

x = -b / 2a

  • Plot the x-intercepts and the vertex on a coordinate plane.
  • Draw a smooth curve through the points to form the graph of the quadratic equation.

Conclusion

Quadratic equations can be a challenging topic, but with practice and patience, you can master the skills needed to solve them. By following the steps outlined in this article, you should be able to tackle even the most complex quadratic equations. Remember to always check your solutions by plugging them back into the original equation to ensure that they are correct.

Tips and Tricks

  • Practice, practice, practice! The more you practice solving quadratic equations, the more comfortable you will become with the process.
  • Use the quadratic formula when the equation cannot be factored easily.
  • Check your solutions by plugging them back into the original equation to ensure that they are correct.
  • Graphing a quadratic equation can help you visualize the solutions and understand the behavior of the equation.

Common Mistakes to Avoid

  • Don't forget to check your solutions by plugging them back into the original equation.
  • Make sure to simplify the equation and isolate the variable x before applying the quadratic formula.
  • Don't confuse the quadratic formula with the formula for the area of a circle (A = πr^2).
  • Be careful when simplifying expressions involving square roots.