Quadratic Equations: Methods of Solution and the Discriminant

Quadratic Equations

Definition and Properties

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where a, b, and c are real numbers and $a \neq 0$. The degree of a quadratic equation is 2, which means the highest exponent of the variable is 2. Quadratic equations can have real or complex (imaginary) solutions.

• Degree: The highest power of the variable (here, 2).

• Complex Solutions: Solutions may be real or imaginary, depending on the equation.

Methods for Solving Quadratic Equations

Solving by Factoring

Factoring is a method where the quadratic is rewritten as a product of two binomials. If one side of the equation is zero, the Zero Factor Property can be applied: if $AB = 0$, then $A = 0$ or $B = 0$.

• Zero Factor Property: If a product equals zero, at least one factor must be zero.

• Factoring Trinomials: Reverse the FOIL method (First, Outer, Inner, Last) and use intuition or trial and error to factor.

Examples:

• Solve $x^2 - x - 12 = 0$

• Solve $(x + 3)(x - 4) = 8$

• Solve $x^2 - 9 = 0$

Solving by the Square Root Property

The Square Root Property is used for equations of the form $x^2 = k$. The solutions are $x = \pm \sqrt{k}$.

• If $k > 0$, there are two real solutions.

• If $k = 0$, there is one real solution ($x = 0$).

• If $k < 0$, there are two imaginary solutions.

Square Root Property:

$\text{If } x^2 = k, \text{ then } x = \pm \sqrt{k}$

Examples:

• Solve $2x^2 - 1 = 9$

• Solve $(x - 3)^2 + 8 = 0$

• Solve $2(x - 4)^2 - 16 = 0$

Solving by Completing the Square

When a quadratic cannot be factored easily, completing the square is used. This method rewrites the equation so one side is a perfect square trinomial, then applies the square root property.

• If the leading coefficient is not 1, divide both sides by it.

• Move the constant term to the other side.

• Find the value to complete the square: take half the coefficient of $x$, square it, and add to both sides.

• Factor the trinomial into a perfect square.

• Solve using the square root property.

Example: Solve $2x^2 - 3x - 4 = 0$ by completing the square.

Solving by the Quadratic Formula

The quadratic formula provides solutions to any quadratic equation $ax^2 + bx + c = 0$ ($a \neq 0$):

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

This formula is derived by completing the square on the general quadratic equation.

Examples:

• Solve $2x^2 + 3x = 2$

• Solve $x^2 = 4x - 9$

The Discriminant

Definition and Use

The discriminant is the expression under the square root in the quadratic formula: $b^2 - 4ac$. It determines the number and type of solutions to a quadratic equation.

Value of $b^2 - 4ac$	Number of Real Solutions	Number of Imaginary Solutions
Positive (> 0)	2	0
Zero (= 0)	1	0
Negative (< 0)	0	2

Example: For $5x^2 - 6x + 2 = 0$, calculate the discriminant to determine the number and type of solutions.

Additional info: The quadratic formula and discriminant are fundamental tools for analyzing and solving quadratic equations, especially when factoring is not straightforward. Mastery of these methods is essential for further study in algebra and calculus.