Solve For $q$. − 25 Q 2 + 81 = − 99 -25 Q^2 + 81 = -99 − 25 Q 2 + 81 = − 99 Write Your Answer In Simplified, Rationalized Form. Q = □ Q = \square Q = □ Or Q = □ Q = \square Q = □

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, $-25 q^2 + 81 = -99$, and provide a step-by-step guide on how to simplify and rationalize the solution.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, q) 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.

Solving the Quadratic Equation

To solve the quadratic equation $-25 q^2 + 81 = -99$, we need to first isolate the variable q. We can do this by subtracting 81 from both sides of the equation:

$$-25 q^2 = -99 - 81$$

$$-25 q^2 = -180$$

Next, we can divide both sides of the equation by -25:

$$q^2 = \frac{-180}{-25}$$

$$q^2 = \frac{180}{25}$$

$$q^2 = \frac{36}{5}$$

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

$$q = \pm \sqrt{\frac{36}{5}}$$

Simplifying the Solution

To simplify the solution, we can rationalize the denominator by multiplying both the numerator and denominator by the square root of the denominator:

$$q = \pm \sqrt{\frac{36}{5}}$$

$$q = \pm \frac{\sqrt{36}}{\sqrt{5}}$$

$$q = \pm \frac{6}{\sqrt{5}}$$

To rationalize the denominator, we can multiply both the numerator and denominator by the square root of 5:

$$q = \pm \frac{6}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$$

$$q = \pm \frac{6\sqrt{5}}{5}$$

Conclusion

In this article, we have solved the quadratic equation $-25 q^2 + 81 = -99$ and simplified the solution to rationalized form. We have shown that the solution is $q = \pm \frac{6\sqrt{5}}{5}$. This solution can be used to solve a variety of problems in mathematics and other fields.

Tips and Tricks

• When solving quadratic equations, it's essential to isolate the variable and simplify the solution.
• Rationalizing the denominator can help simplify the solution and make it easier to work with.
• When working with square roots, it's essential to remember that the square root of a number is always positive, unless specified otherwise.

Common Quadratic Equations

• $$x^2 + 4x + 4 = 0$$

• $$x^2 - 6x + 9 = 0$$

• $$x^2 + 2x + 1 = 0$$

Solving Quadratic Equations with Complex Solutions

In some cases, quadratic equations may have complex solutions. A complex solution is a solution that involves the imaginary unit, i, which is defined as the square root of -1.

• $$x^2 + 1 = 0$$

• $$x^2 - 4 = 0$$

• $$x^2 + 3x + 2 = 0$$

Conclusion

Frequently Asked Questions

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, q) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the variable and simplify the solution. Here are the steps:

1. Isolate the variable by subtracting the constant term from both sides of the equation.
2. Divide both sides of the equation by the coefficient of the squared term.
3. Take the square root of both sides of the equation.
4. Simplify the solution by rationalizing the denominator.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (in this case, q) is one. The general form of a linear equation is:

$$ax + b = 0$$

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. A complex solution is a solution that involves the imaginary unit, i, which is defined as the square root of -1.

Q: How do I determine if a quadratic equation has complex solutions?

A: To determine if a quadratic equation has complex solutions, you need to check the discriminant. The discriminant is the expression under the square root in the quadratic formula:

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

If the discriminant is negative, then the quadratic equation has complex solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is:

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

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, you can use the quadratic formula to solve any quadratic equation. However, you need to make sure that the equation is in the form:

$$ax^2 + bx + c = 0$$

Q: What are some common quadratic equations?

A: Some common quadratic equations include:

• $$x^2 + 4x + 4 = 0$$

• $$x^2 - 6x + 9 = 0$$

• $$x^2 + 2x + 1 = 0$$

Q: How do I graph a quadratic equation?

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

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

A: Quadratic equations have many 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 bridges and other structures.
• Economics: Quadratic equations are used to model the behavior of economic systems.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics and have many real-world applications. By understanding how to solve quadratic equations and using the quadratic formula, you can solve a wide range of problems in mathematics and other fields.