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Quadratic Equations and Problem Solving

Quadratic Equations and Problem Solving

We now focus on solving quadratic equations. Recall we learned how to solve for linear equations
(section 2-3); we will also solve quadratic equations graphically, numerically, and symbolically.

Note -> a quadratic equation can have zero, one, or two real solutions.

Quadratic Equation: A quadratic equation in one variable is an equation that can be written in the
form ax^2 + bx + c = 0 , where a, b, and c are real numbers with a ≠ 0.

Solving Quadratic Equations

4 basic symbolic strategies:

Factoring, Square Root Property, Completing the Square, Quadratic Formula
Factoring -> based on the zero-product property, if ab = 0 , then a = 0 or b = 0 or both.

Ex: x^2 - 4 = 0

Square Root Property -> Let k be a nonnegative number. Then the solutions to the equation

Ex: x^2 - 4 = 0 ->

Completing the Square -> useful when solving quadratic equations that do not factor easily!
Recall from 3-1 that the equation can be solved using x^2 + kx + (k / 2)^2 = (x + k / 2)^2 for k constant.
Ex:

Quadratic Formula -> solutions to the quadratic equation ax^2 + bx + c = 0 , where a ≠ 0, are

Ex:

Discriminant: In the quadratic formula, the quantity b^2 – 4ac is called the discriminant. The
discriminant provides the number of real solutions to a quadratic equation.

Note -> When a quadratic equation is solved graphically, the parabola intersects the x-axis 0, 1,
or 2 times. Where it crosses x-axis is (are) the x-intercepts!

• If b^2 – 4ac > 0, there are 2 real solutions
• If b^2 – 4ac = 0, there is 1 real solution
• If b^2 – 4ac < 0, there are no real solutions (these may be complex numbers)

GRAPHICALLY
Graph the quadratic equation and locate the x-intercepts.
Ex: x^2 - 4 = 0

NUMERICALLY
Make a table for quadratic equation y, Wherever y = 0 then these are “the zeros of y”, the solutions
(recall x-intercepts)!
Ex: x^2 - 4 = 0

Problem Solving

Exs: