QUADRATIC EQUATIONS A general quadratic equation can be written in the form ax bx c 0. A quadratic equation has two solutions, called roots. These two solutions, or roots, may or may not be distinct, and they may or may not be real. The number of real roots of a quadratic equation ax bx c 0 correspond to the number of x- intercepts of the graph of the related quadratic function f ( x) ax bx c. Since the graph of a quadratic function can have zero, one, or two x-intercepts (called the zeroes of the quadratic function), the related quadratic equation will have zero, one, or two real roots. These three cases are illustrated as follows: The methods that we will use to solve quadratic equations include: Factoring Completing the Square Quadratic Formula Before we solve quadratic equations by factoring, we will first review various methods of factoring quadratic expressions.
FACTORING QUADRATIC EXPRESSIONS 1. Common Factor: Factor each of the following expressions. a. 3x 6 x b. 5 x( x ) 3( x ) c. 8x 6x 4x 3 If the terms of a polynomial expression contain a common factor, always factor it out first.. Trinomials of the form x bx c Factor a trinomial of the form ax bx c; a 1 as follows: Factor each of the following expressions. a. x 5x 4 b. x 7xy 18y c. 5t 5t 30 3. Trinomials of the form ax bx c, where a 1: Factor each of the following expressions a. 3x 11x 4 b. 6x 13xy 6y c. 4x 30x 9
4. Difference of Squaresax bx : Factor a difference of squares as follows: ( ax) ( by) = ( ax by)( ax by) Factor each of the following expressions. 4 a. x 36 b. 16 9 x y c. 7x 1 5. Perfect Trinomial Squares: a ab b or a ab b a ab b a b Factor a perfect square trinomial as follows: Factor each of the following expressions. a ab b a b or a. 9x 30x 5 b. 16x 8xy y Factoring Polynomials Having a Quadratic Pattern: A. We can factor a polynomial in quadratic form, follows: a( P) b( P) c, where P is any expression, as Replace the expression P with a temporary variable, say k Factor as usual Replace k with the expression P and simplify Factor the following examples: a. ( x 3) ( x 3) 0 b. 1( a a 5) 13( a a 5) 4
c. 1( m ) 4( m ) 9 B. We can factor a polynomial in the form of a difference of squares, ( P Q)( P Q) where P and Q are any expressions. P Q, as Factor the following examples: a. 9(m 1) 4( n ) 1 3 16 5 3 b. a b
SOLVING QUADRATIC EQUATIONS BY FACTORING Some quadratic equations that have real-number solutions can be factored easily. The zero product property states that if the product of two or more numbers equals zero, then at least one of the numbers must be zero. For example, if A B = 0, then A and/or B equals zero. To solve a quadratic equation of the form ax bx c 0 by factoring, factor the quadratic expression and set each factor equal to zero and solve for the roots. Example 1: Solve Quadratic Equations by Factoring Determine the roots of the following quadratic equations: a. x 10x 4 0 b. 0.15x 0.875x 1.5 c. x 9x 5
d. 5 x x 3 3 e. 9x 4x 49 f. 18x 7 x g. 8x 50 0
Example : Apply Quadratic Equations Dock jumping is an event in which dogs compete for the longest jumping distance from a dock into a body of water. The path of a dog on a particular jump can be approximated by the 3 11 quadratic function h( d) d d, where 10 10 h is the height above the surface of the water and d is the horizontal distance the dog travels from the edge of the dock, both in feet. Determine the horizontal distance of the jump. Solution: When the dog lands in the water, the dog s height above surface of the water is 0 feet, so set h = 0 and solve for d.
Example 3: Write and Solve a Quadratic Equation A right triangle has a perimeter of 56 cm. If the length of the hypotenuse is 5 cm, determine the lengths of the other two sides. Solution:
Example 4: Write and Solve a Quadratic Equation A 60 m by 40 m factory is to be built on a rectangular lot. A lawn of uniform width, equal to the area of the factory, must surround it. a. How wide is the strip of lawn? b. What are the dimensions of the lot? Solution:
ALTERNATIVE VERSION Example 4: Write and Solve a Quadratic Equation A factory is to be built on a lot that measures 80 m by 60 m. A lawn of uniform width, equal to the area of the factory, must surround it. c. How wide is the strip of lawn? d. What are the dimensions of the factory? Solution:
Example 5: Write and Solve a Quadratic Equation An open-topped box is to be made from a rectangular piece of tin measuring 50 cm by 40 cm by cutting squares of equal size from each corner. The base area is to be 875 square centimetres. What is the volume of the box?