Lesson 9MA 152, Section 1.3 & 1.4 (part 1)
A quadratic equation is a 2nd degree polynomial equation of the form .
Quadratic equations can be solved by four different methods.
- Using factoring and the zero-factor theorem
- Using the square root property
- Completing the square
- Using the quadratic formula
In this lesson, we will cover all the methods above and some application problems.
Zero-Factor Theorem: If a and b are real numbers, and if
If a quadratic equation is written in standard form (descending order equal to zero) and is factorable, the zero-factor theorem can be used to solve the equation.
Ex 1: Solve by using factoring.
Square Root Property: If .
Ex 2: Use the square root property to solve each equation.
Remember a perfect square trinomial has the form . Notice that half of the middle coefficient square equals the last term. If the leading coefficient is a 1, every binomial of the form can be made into a perfect square trinomial by adding . If this value is added to both sides of an equation of the form , then the square root property can be used to solve the equation. This process is called the completing the square process.
Here is an example:
These are the steps that were used:
- Use this procedure only on equations where the leading coefficient is 1.
- Move the constant to the other side so the equation is in the form .
- Complete the square by taking half of the coefficient of x, squaring it, and adding to both sides of the equation.
- Factor the perfect square trinomial.
- Solve by using the square root property.
Ex 3: Solve by completing the square.
Not all quadratic equations can be solved using factoring and the zero-factor theorem or the square root property. Not all quadratic equations can easily be solved by completing the square. There is a 4th method that could always be used to solve a quadratic function, the quadratic formula. The quadratic formula is actually derived by the completing the square process. We will not discuss this derivation process in class, but it is in the textbook, if you want to see how.
Quadratic Formula: The solutions of the general quadratic equation, , are .
Ex 1: Solve by using the quadratic formula.
The part of the quadratic formula found under the square root, , is known as the discriminant. It can be used to determine how many roots (solutions) and the nature of the solution(s).
Value of / Number of Roots / Nature of Roots0 / 1 / rational
positive perfect square / 2 / rational
positive nonperfect square / 2 / irrational
negative / not real
Ex 2: Use the discriminant to determine the nature of the roots of each equation.
Sometimes a rational equation can be written as a quadratic equation and solved by any of the four methods covered. Remember, no solutions can make zero denominators.
Ex 3: Solve each equation by an appropriate method.
Application of Quadratic Equations: We will begin some application problems in this lesson that are translated as quadratic equation. These types of problems will be continued in the next lesson.
We well begin with some geometric problems.
1.The side of a square is 4 centimeters shorter than the side of a second square. If the sum of their areas is 106 square centimeters, find the length of one side of the larger square.
2.A large permanent movie screen is in the Panasonic Imax Theater in Sydney, Australia. The rectangular screen has an area of 11,349 square feet. Find the dimensions of the screen if it is 20 feet longer than it is wide.
3.A piece of thick cardstock, 20 inches per side, is to have four equal squares cut from its corners as shown. If the edges are then to be folded up to make an open topped box with a bottom area of 256 square inches, find the depth of the box.