Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. These equations are characterized by their parabolic graphs, which can open upwards or downwards depending on the sign of a. The solutions to quadratic equations can be found through various methods, including factoring, completing the square, and using the quadratic formula.

Factoring Quadratic Equations

Factoring involves expressing the quadratic equation in the form of (px + q)(rx + s) = 0. To factor a quadratic equation, follow these steps:

1. Identify the coefficients a, b, and c.
2. Find two numbers that multiply to ac and add up to b.
3. Rewrite the middle term using the two numbers found.
4. Factor by grouping.
5. Set each factor equal to zero and solve for x.

Example: Solve x² + 5x + 6 = 0.

The numbers 2 and 3 multiply to 6 and add to 5. Thus, we factor to (x + 2)(x + 3) = 0, giving solutions x = -2 and x = -3.

Completing the Square

This method involves manipulating the quadratic equation into a perfect square trinomial. Steps include:

1. Start with the standard form ax² + bx + c = 0.
2. If a is not 1, divide the entire equation by a.
3. Move c to the other side of the equation.
4. Take b, halve it, square it, and add it to both sides.
5. Factor the left side as a squared binomial.
6. Solve for x by taking the square root of both sides.

Example: Solve x² + 6x + 5 = 0.

Rewrite as x² + 6x = -5. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 4, which factors to (x + 3)² = 4. Thus, x + 3 = ±2, giving solutions x = -1 and x = -5.

The Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides a direct method to find the roots of any quadratic equation. The discriminant b² - 4ac indicates the nature of the roots:

• If > 0: two distinct real roots.
• If = 0: one real root (double root).
• If < 0: no real roots (two complex roots).

Example: Solve 2x² + 4x - 6 = 0 using the quadratic formula.

Here, a = 2, b = 4, c = -6. Calculate discriminant: 4² - 4(2)(-6) = 16 + 48 = 64, which is > 0. Now apply the formula:

x = (-4 ± √64) / (2*2) = (-4 ± 8) / 4. Solutions: x = 1 and x = -3.

Maximum and Minimum Values

The vertex of a quadratic function y = ax² + bx + c provides the maximum or minimum value. The vertex occurs at x = -b/(2a). If a > 0, the parabola opens upwards, and the vertex represents the minimum value. If a < 0, the parabola opens downwards, and the vertex represents the maximum value.

Graphs

Graphs of quadratic equations are parabolas. Key features include:

• Vertex: The highest or lowest point.
• Axis of Symmetry: Vertical line through the vertex, x = -b/(2a).
• X-intercepts (roots): Points where the graph crosses the x-axis, found using any of the solving methods.
• Y-intercept: Point where the graph crosses the y-axis, found by evaluating f(0).

Example: For y = x² - 4x + 3, the vertex is at x = 2, with y-value -1, and roots at x = 1 and x = 3.