Ax Bx C Solve For X

Solving for x: A Comprehensive Guide to Quadratic Equations of the Form ax² + bx + c = 0

The seemingly simple equation, ax² + bx + c = 0, represents a cornerstone of algebra – the quadratic equation. Understanding how to solve for 'x' in this equation is crucial for a wide range of applications in mathematics, science, and engineering. This comprehensive guide will explore various methods for solving quadratic equations, providing detailed explanations and examples to solidify your understanding. We'll delve into the quadratic formula, factoring, completing the square, and examine the nature of the solutions (real or complex).

Understanding the Quadratic Equation

Before diving into the solution methods, let's define the terms in the equation ax² + bx + c = 0:

• a, b, and c: These are constants, or numbers, with 'a' not equal to zero (a ≠ 0). If 'a' were zero, the equation would become linear, not quadratic.
• x: This is the variable we aim to solve for. It represents the unknown value(s) that satisfy the equation.
• ax²: This term is the quadratic term, representing the highest power of x.
• bx: This is the linear term, representing x raised to the power of one.
• c: This is the constant term, a numerical value without any 'x'.

Method 1: Factoring

Factoring is a method used to find the solutions (roots) of a quadratic equation by expressing it as a product of two linear expressions. This method is particularly efficient when the quadratic equation has easily identifiable factors.

Steps:

1. Rewrite the equation: Ensure the equation is in the standard form: ax² + bx + c = 0.
2. Factor the quadratic expression: Find two binomials whose product equals the original quadratic expression. This often involves finding two numbers that add up to 'b' and multiply to 'ac'.
3. Set each factor equal to zero: Once factored, set each of the linear expressions to zero and solve for 'x'.

Example:

Solve for x in the equation x² + 5x + 6 = 0.

1. Rewrite: The equation is already in standard form.
2. Factor: We need two numbers that add to 5 (b) and multiply to 6 (ac). These numbers are 2 and 3. Therefore, the factored form is (x + 2)(x + 3) = 0.
3. Set factors to zero:
   • x + 2 = 0 => x = -2
   • x + 3 = 0 => x = -3

Therefore, the solutions are x = -2 and x = -3. We can verify these solutions by substituting them back into the original equation.

Method 2: Completing the Square

Completing the square is a powerful technique that can be used to solve any quadratic equation, even those that are not easily factored. This method involves manipulating the equation to create a perfect square trinomial.

Steps:

1. Ensure 'a' is 1: Divide the entire equation by 'a' if it's not already 1.
2. Move the constant term to the right side: Move the constant term ('c') to the right side of the equation.
3. Find the value to complete the square: Take half of the coefficient of the linear term (b/2), square it ((b/2)²), and add it to both sides of the equation.
4. Factor the perfect square trinomial: The left side of the equation will now be a perfect square trinomial, which can be factored into (x + b/2)².
5. Solve for x: Take the square root of both sides of the equation and solve for 'x'. Remember to consider both positive and negative square roots.

Example:

Solve for x in the equation 2x² + 8x - 10 = 0.

1. 'a' is not 1: Divide the entire equation by 2: x² + 4x - 5 = 0.
2. Move constant: x² + 4x = 5.
3. Complete the square: Half of 4 is 2, and 2² is 4. Add 4 to both sides: x² + 4x + 4 = 9.
4. Factor: (x + 2)² = 9.
5. Solve:
   • x + 2 = ±3
   • x = -2 ± 3
   • x = 1 or x = -5

Therefore, the solutions are x = 1 and x = -5.

Method 3: The Quadratic Formula

The quadratic formula is a universal method that provides a direct solution for 'x' in any quadratic equation, regardless of its factorability. This formula is derived from completing the square and is extremely valuable for solving complex equations.

The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

Steps:

1. Identify a, b, and c: Identify the values of a, b, and c from the quadratic equation in standard form (ax² + bx + c = 0).
2. Substitute into the formula: Substitute the values of a, b, and c into the quadratic formula.
3. Calculate: Perform the calculations to find the values of x.

Example:

Solve for x in the equation 3x² - 7x + 2 = 0.

1. Identify a, b, c: a = 3, b = -7, c = 2.
2. Substitute: x = [7 ± √((-7)² - 4 * 3 * 2)] / (2 * 3) x = [7 ± √(49 - 24)] / 6 x = [7 ± √25] / 6 x = [7 ± 5] / 6
3. Calculate:
   • x = (7 + 5) / 6 = 2
   • x = (7 - 5) / 6 = 1/3

Therefore, the solutions are x = 2 and x = 1/3.

The Discriminant (b² - 4ac)

The expression within the square root in the quadratic formula, b² - 4ac, is called the discriminant. The discriminant provides valuable information about the nature of the solutions:

• b² - 4ac > 0: The equation has two distinct real roots.
• b² - 4ac = 0: The equation has one real root (a repeated root).
• b² - 4ac < 0: The equation has two complex roots (involving imaginary numbers, 'i', where i² = -1).

Applications of Quadratic Equations

Quadratic equations are far from mere mathematical exercises. They find extensive application in various fields:

• Physics: Calculating projectile motion, determining the path of a thrown object, and analyzing oscillatory systems.
• Engineering: Designing bridges, analyzing stress and strain in structures, and optimizing circuit designs.
• Economics: Modeling supply and demand curves, predicting market trends, and optimizing profit functions.
• Computer graphics: Creating curves and shapes in two and three dimensions.
• Finance: Calculating compound interest and modeling investment growth.

Choosing the Right Method

The best method for solving a quadratic equation depends on the specific equation and your comfort level with each technique. Factoring is efficient for easily factorable equations. Completing the square offers a systematic approach, while the quadratic formula provides a universal solution. Understanding the discriminant allows you to predict the nature of the solutions before performing any calculations.

Conclusion

Solving for 'x' in the quadratic equation ax² + bx + c = 0 is a fundamental skill in algebra. Mastering the various methods – factoring, completing the square, and using the quadratic formula – along with understanding the discriminant, equips you with a versatile toolset for tackling a vast array of mathematical problems across numerous disciplines. Consistent practice and a firm grasp of these techniques will lay a strong foundation for your further mathematical pursuits. Remember to always verify your solutions by substituting them back into the original equation.