Topic 4: Quadratic Equation (Straighterline)
Remembering The Formula
"Pop Goes the Weasel": "x is equal to minus b plus or minus the square root of b-squared minus four a c ALL over two a" ♫ "All around the mulberry bush The monkey chased the weasel. The monkey thought 'twas all in fun. Pop goes the weasel
Quadratic Equation:
A Quadratic Equation (x exponent makes it quadratic) 5x^2 - 3x + 3 = 0 The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2). It is also called an "Equation of Degree 2" (because of the "2" on the x)
Imaginary Numbers
An Imaginary Number, when squared, gives a negative result. In mathematics we use i (for imaginary)
The Quadratic Formula
For ax2 + bx + c = 0, the values of x which are the solutions of the equation are given by:
the zero-product property
How do we know there are no more solutions other than the two we find using our method? The answer is provided by a simple but very useful property, called the zero-product property: If the product of two quantities is equal to zero, then at least one of the quantities must be equal to zero.
Summary of Quadratic Equation
Quadratic Equation in Standard Form: ax2 + bx + c = 0 Quadratic Equations can be factored Quadratic Formula: x = −b ± √(b2 − 4ac)/2a When the Discriminant (b2−4ac) is: positive, there are 2 real solutions zero, there is one real solution negative, there are 2 complex solutions
Solving by factoring
Suppose we want to solve the equation x^2-3x-10=0, we have to factor then solve as normal. Since the leading coefficient—the coefficient of x^2—is 1, we can use the sum-product pattern. According to the sum-product pattern, if we find two numbers a and b such that a+b=-3 and a* b= -10 then x ^2 -3x -10 =(x+a)(x+b). By considering the pairs of integers whose product is -10 then eliminating those whose sum isn't -3 we find that the numbers we need are a=2 b=-5. Therefore, the factored expression is (x+2)(x-5) (x +2) (x-5) = 0
Standard Form
The Standard Form of a Quadratic Equation looks like this: Quadratic Equation: ax^2 + bx + c = 0 a, b and c are known values. a can't be 0. "x" is the variable or unknown (we don't know it yet).
Solving factored quadratic equations
This is a product of two expressions that is equal to zero. Note that any x value that makes either (x-1)(x−1) zero, will make their product zero. (x-1)(x+3)=0 x-1=0 x=1 or x+3=0 x=-3 Substituting either x=1 x=-3 into the equation will result in the true statement 0=0 so they are both solutions to the equation.
Complex Solutions to Quadratic Equations
When the Discriminant (the value b2 − 4ac) is negative we get a pair of Complex solutions ... what does that mean? It means our answer will include Imaginary Numbers.
Discriminant
b2 − 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer: when b2 − 4ac is positive, we get two Real solutions when it is zero we get just ONE real solution (both answers are the same) when it is negative we get a pair of Complex solutions