Solving 10x^2 + 9 = X A Step-by-Step Guide
In this article, we will delve into the intricacies of solving the quadratic equation 10x² + 9 = x. Quadratic equations, characterized by their highest power being two, are fundamental in mathematics and find applications in various fields like physics, engineering, and economics. Mastering the techniques to solve them is crucial for anyone pursuing studies or careers in these areas. This guide will provide a step-by-step approach to solving the given equation, explore different methods, and discuss the nature of the solutions.
Understanding Quadratic Equations
Before we dive into the specifics of the equation 10x² + 9 = x, let's establish a solid understanding of what quadratic equations are and their general form. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is two. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The coefficients a, b, and c play a significant role in determining the nature and values of the solutions to the equation.
In the equation ax² + bx + c = 0, a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. The solutions to the quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the parabola represented by the quadratic equation intersects the x-axis. Quadratic equations can have two distinct real solutions, one repeated real solution, or two complex solutions, depending on the discriminant.
Transforming the Equation into Standard Form
Our given equation is 10x² + 9 = x. To solve it effectively, the first step is to rewrite it in the standard quadratic form ax² + bx + c = 0. This involves rearranging the terms so that all terms are on one side of the equation, and the other side is zero. Subtracting x from both sides of the equation, we get:
10x² - x + 9 = 0
Now, the equation is in the standard quadratic form, where a = 10, b = -1, and c = 9. With the equation in this form, we can proceed to solve it using various methods.
Methods for Solving Quadratic Equations
There are several methods to solve quadratic equations, each with its strengths and weaknesses. The most common methods are factoring, completing the square, and using the quadratic formula. We will explore each method and determine the most suitable approach for our equation, 10x² - x + 9 = 0.
1. Factoring
Factoring involves expressing the quadratic equation as a product of two linear factors. This method is efficient when the quadratic expression can be easily factored. However, not all quadratic equations are factorable using integers. To factor the equation 10x² - x + 9 = 0, we need to find two binomials that, when multiplied, give us the original quadratic expression. This involves finding two numbers that multiply to give the product of a and c (10 * 9 = 90) and add up to b (-1). In this case, it's difficult to find such integer pairs, suggesting that factoring might not be the most straightforward method for this particular equation.
2. Completing the Square
Completing the square is a method that transforms the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. This method is particularly useful when the quadratic equation is not easily factorable. To complete the square for 10x² - x + 9 = 0, we first divide the entire equation by the leading coefficient (10) to make the coefficient of x² equal to 1:
x² - (1/10)x + 9/10 = 0
Next, we move the constant term to the right side of the equation:
x² - (1/10)x = -9/10
Now, we add the square of half of the coefficient of x to both sides. Half of the coefficient of x is -(1/20), and its square is (1/400). Adding this to both sides, we get:
x² - (1/10)x + 1/400 = -9/10 + 1/400
The left side is now a perfect square trinomial, which can be written as:
(x - 1/20)² = -359/400
Since the right side is negative, taking the square root will result in complex solutions. This indicates that the equation has no real roots, but we can still find the complex solutions. Taking the square root of both sides, we get:
x - 1/20 = ±√(-359/400)
x - 1/20 = ±(i√359)/20
x = 1/20 ± (i√359)/20
Thus, the solutions are complex numbers.
3. Quadratic Formula
The quadratic formula is a general formula that provides the solutions to any quadratic equation in the form ax² + bx + c = 0. The formula is given by:
x = [-b ± √(b² - 4ac)] / (2a)
This method is highly reliable and works for all quadratic equations, regardless of whether they are factorable or not. For our equation, 10x² - x + 9 = 0, we have a = 10, b = -1, and c = 9. Plugging these values into the quadratic formula, we get:
x = [-(-1) ± √((-1)² - 4 * 10 * 9)] / (2 * 10)
x = [1 ± √(1 - 360)] / 20
x = [1 ± √(-359)] / 20
x = [1 ± i√359] / 20
x = 1/20 ± (i√359)/20
As we obtained with completing the square, the solutions are complex numbers.
Analyzing the Solutions
From both the method of completing the square and the quadratic formula, we obtained the solutions:
x = 1/20 ± (i√359)/20
These solutions are complex conjugates, meaning they have the form p + qi and p - qi, where p and q are real numbers, and i is the imaginary unit (√-1). This indicates that the quadratic equation 10x² - x + 9 = 0 has no real roots, and the parabola represented by the equation does not intersect the x-axis.
The Discriminant
The nature of the solutions to a quadratic equation can also be determined by examining the discriminant, which is the part of the quadratic formula under the square root sign: b² - 4ac. The discriminant provides valuable information about the number and type of solutions:
- If b² - 4ac > 0, the equation has two distinct real solutions.
- If b² - 4ac = 0, the equation has one repeated real solution.
- If b² - 4ac < 0, the equation has two complex solutions.
For our equation, 10x² - x + 9 = 0, the discriminant is:
(-1)² - 4 * 10 * 9 = 1 - 360 = -359
Since the discriminant is negative (-359 < 0), this confirms that the equation has two complex solutions, which aligns with our previous results.
Conclusion
In conclusion, the quadratic equation 10x² + 9 = x, when rewritten in standard form as 10x² - x + 9 = 0, has two complex solutions:
x = 1/20 + (i√359)/20 and x = 1/20 - (i√359)/20
We successfully solved the equation using the methods of completing the square and the quadratic formula. Additionally, analyzing the discriminant confirmed that the equation has complex solutions. Understanding these methods and the nature of solutions is crucial for solving quadratic equations and applying them in various mathematical and real-world contexts. This comprehensive guide provides a clear understanding of the steps involved and the underlying principles, enabling you to tackle similar problems with confidence.
