Solve The Equation: 9 X 2 − 6 X + 50 = 0 9x^2 - 6x + 50 = 0 9 X 2 − 6 X + 50 = 0

Mar 9, 2025 by ADMIN 81 views

**Solving Quadratic Equations: A Step-by-Step Guide to Finding the Roots of $9x^2 - 6x + 50 = 0$**

Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. 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. In this article, we will focus on solving the quadratic equation $9x^2 - 6x + 50 = 0$ .

Understanding the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where a, b, and c are the coefficients of the quadratic equation. To use the quadratic formula, we need to identify the values of a, b, and c in the given equation.

Identifying the Coefficients

In the equation $9x^2 - 6x + 50 = 0$ , we can identify the coefficients as follows:

$a = 9$
$b = -6$
$c = 50$

Applying the Quadratic Formula

Now that we have identified the coefficients, we can apply the quadratic formula to find the roots of the equation.

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(9)(50)}}{2(9)}$

$x = \frac{6 \pm \sqrt{36 - 1800}}{18}$

$x = \frac{6 \pm \sqrt{-1764}}{18}$

$x = \frac{6 \pm 42i}{18}$

$x = \frac{1 \pm 7i}{3}$

Interpreting the Results

The quadratic formula has given us two complex roots: $x = \frac{1 + 7i}{3}$ and $x = \frac{1 - 7i}{3}$ . These roots are complex numbers, which means they have both real and imaginary parts.

Understanding Complex Numbers

Complex numbers are numbers that have both real and imaginary parts. They are denoted by the symbol $a + bi$ , where a is the real part and b is the imaginary part. In the case of the roots we found, the real part is $\frac{1}{3}$ and the imaginary part is $\pm \frac{7}{3}$ .

Conclusion

Solving the quadratic equation $9x^2 - 6x + 50 = 0$ using the quadratic formula has given us two complex roots. These roots are $\frac{1 + 7i}{3}$ and $\frac{1 - 7i}{3}$ . Complex numbers are an essential concept in mathematics, and they have many applications in various fields.

Real-World Applications of Quadratic Equations

Quadratic equations have many real-world applications, including:

Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Tips for Solving Quadratic Equations

Here are some tips for solving quadratic equations:

Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
Identify the coefficients: Make sure to identify the coefficients a, b, and c in the given equation.
Check for complex roots: Quadratic equations can have complex roots, so make sure to check for them.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving quadratic equations:

Not identifying the coefficients: Make sure to identify the coefficients a, b, and c in the given equation.
Not using the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
Not checking for complex roots: Quadratic equations can have complex roots, so make sure to check for them.

Conclusion

Solving quadratic equations is an essential skill in mathematics, and it has many real-world applications. By using the quadratic formula and identifying the coefficients, we can find the roots of a quadratic equation. Remember to check for complex roots and avoid common mistakes. With practice and patience, you can become proficient in solving quadratic equations.

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 our previous article, we discussed how to solve quadratic equations using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations.

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.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

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

Q: How do I identify the coefficients in a quadratic equation?

A: To identify the coefficients in a quadratic equation, you need to look at the equation and identify the values of a, b, and c. For example, in the equation $9x^2 - 6x + 50 = 0$ , the coefficients are:

$a = 9$
$b = -6$
$c = 50$

Q: What is the difference between a real root and a complex root?

A: A real root is a root that is a real number, whereas a complex root is a root that is a complex number. Complex numbers have both real and imaginary parts.

Q: How do I determine if a quadratic equation has real or complex roots?

A: To determine if a quadratic equation has real or complex roots, you need to look at the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It is given by:

$b^2 - 4ac$

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to:

Identify the coefficients a, b, and c in the equation.
Plug the values of a, b, and c into the quadratic formula.
Simplify the expression under the square root.
Solve for x.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not identifying the coefficients a, b, and c in the equation.
Not using the quadratic formula.
Not checking for complex roots.
Not simplifying the expression under the square root.

Q: How do I check if a quadratic equation has complex roots?

A: To check if a quadratic equation has complex roots, you need to look at the discriminant. If the discriminant is negative, the equation has two complex roots.

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 describe the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. By understanding the quadratic formula and identifying the coefficients, we can solve quadratic equations and determine if they have real or complex roots. Remember to check for complex roots and avoid common mistakes. With practice and patience, you can become proficient in solving quadratic equations.