Solving Quadratic Equations: Study Notes and Examples

Quadratic Functions and Equations

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two, generally written in the standard form:

• Standard Form: $ax^2 + bx + c = 0$, where $a \neq 0$

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.

Factoring Method

• Express the quadratic equation in the form $ax^2 + bx + c = 0$.

• Factor the quadratic expression, if possible.

• Set each factor equal to zero and solve for $x$.

Quadratic Formula

If factoring is not possible, use the quadratic formula:

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

• Discriminant: $b^2 - 4ac$ determines the nature of the roots.

Examples

• Example 1: Solve $x^2 = 9$ Solution:

  • Rewrite as $x^2 - 9 = 0$

  • Factor: $(x - 3)(x + 3) = 0$

  • Set each factor to zero: $x - 3 = 0$ or $x + 3 = 0$

  • Solutions: $x = 3$ or $x = -3$

• Example 2: Solve $x^2 = 49$ Solution:

  • Rewrite as $x^2 - 49 = 0$

  • Factor: $(x - 7)(x + 7) = 0$

  • Set each factor to zero: $x - 7 = 0$ or $x + 7 = 0$

  • Solutions: $x = 7$ or $x = -7$

Key Properties

• Quadratic equations can have two real solutions, one real solution, or two complex solutions depending on the discriminant.

• Factoring is possible when the quadratic can be written as a product of two binomials.

Table: Nature of Roots Based on Discriminant

Discriminant ($b^2 - 4ac$)	Nature of Roots
> 0	Two distinct real roots
= 0	One real root (repeated)
< 0	Two complex roots

Additional info: These examples and explanations cover the basic methods for solving quadratic equations, which are foundational for further study in trigonometry and algebra.