Quadratic Equations: Solving Methods

• A quadratic equation is a polynomial equation of the second degree.
• The general form is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Quadratic equations can be solved by factoring, completing the square, using the quadratic formula, or graphing.

Factoring

• Factoring involves rewriting the quadratic expression as a product of two binomials.
• For example, x^2 + 5x + 6 = (x + 2)(x + 3).
• Setting each factor equal to zero gives the solutions: x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3.
• Factoring is most straightforward when the roots are integers or simple fractions.

Completing the Square

• Completing the square involves manipulating the quadratic equation to form a perfect square trinomial.
• Start by isolating the x^2 and x terms: ax^2 + bx = -c.
• If a ≠ 1, divide the entire equation by a to get x^2 + (b/a)x = -c/a.
• Add (b/2a)^2 to both sides of the equation to complete the square: x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2.
• Rewrite the left side as a squared binomial: (x + b/2a)^2 = -c/a + (b/2a)^2.
• Take the square root of both sides and solve for x.

Quadratic Formula

• The quadratic formula provides a direct method for finding the solutions of any quadratic equation.
• Given ax^2 + bx + c = 0, the solutions are given by x = (-b ± √(b^2 - 4ac)) / (2a).
• The term inside the square root, b^2 - 4ac, is called the discriminant.

The Discriminant

• The discriminant, denoted as Δ = b^2 - 4ac, determines the nature of the roots of the quadratic equation.
• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has one real root (a repeated or double root).
• If Δ < 0, the equation has two complex conjugate roots.

Nature of Roots

• When Δ > 0, the two distinct real roots are x1 = (-b + √Δ) / (2a) and x2 = (-b - √Δ) / (2a).
• When Δ = 0, the single real root is x = -b / (2a).
• When Δ < 0, the two complex conjugate roots are x = (-b ± i√|Δ|) / (2a), where i is the imaginary unit (i^2 = -1).

Examples

• Example 1: x^2 - 4x + 4 = 0. Here, a = 1, b = -4, and c = 4. Δ = (-4)^2 - 4(1)(4) = 16 - 16 = 0. There is one real root: x = -(-4) / (2*1) = 2.
• Example 2: x^2 - 5x + 6 = 0. Here, a = 1, b = -5, and c = 6. Δ = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Since Δ > 0, there are two distinct real roots: x = (5 ± √1) / 2 = (5 ± 1) / 2, so x = 3 or x = 2.
• Example 3: x^2 + x + 1 = 0. Here, a = 1, b = 1, and c = 1. Δ = (1)^2 - 4(1)(1) = 1 - 4 = -3. Since Δ < 0, there are two complex conjugate roots: x = (-1 ± i√3) / 2.

Graphing Quadratic Equations

• The graph of a quadratic equation y = ax^2 + bx + c is a parabola.
• The vertex of the parabola is at x = -b / (2a).
• The x-intercepts (roots) are the solutions to the equation ax^2 + bx + c = 0.
• If the discriminant is positive, the parabola intersects the x-axis at two points.
• If the discriminant is zero, the parabola touches the x-axis at one point (the vertex).
• If the discriminant is negative, the parabola does not intersect the x-axis.

Applications

• Quadratic equations are used in various fields such as physics, engineering, and economics.
• They can model projectile motion, optimization problems, and growth/decay scenarios.
• Understanding the discriminant can provide insights into the nature of solutions without explicitly solving the equation.

Learn about quadratic equations, their general form, and various methods to solve them. This includes factoring, completing the square, and other techniques. Understand how to apply these methods to find the solutions of quadratic equations.