Warm-Up How can you determine whether two lines are parallel or perpendicular or neither? 4

Section 1-6 Solving Quadratic Equations

Quadratic Equations Definition: Any equation that can be written in the form Where a ≠0 (For if a = 0, the equation becomes a linear equation.), is called a quadratic equation. ( Note: The letters a, b, and c are called coefficients) Note: the actual function is written as Definition: A root or solution, of a quadratic equation is a value of the variable that satisfies the equation (i.e. what x value gives me zero?) 3 ways to solve: (1) Factoring (2) Completing the square (3) the quadratic formula y = x2 − x − 2

Factoring Recall: want to solve IDEA - rewrite this equation as the product of two factors ( )*( )=0 (this is called factoring) Now if at least one factor is zero, the equation is true Note: the quadratic equation must be written in standard form before it can be solved by factoring.

Factoring Tips When trying to factor, that is in the form ( )*( )=0, multiply a*c and write out the factors – these factors must add to give b. 6*1=6 (-6)(-1)=6 2*3=6 (-2)(-3)=6

Completing the Square Recall: want to solve IDEA: make one side a perfect square (that is (x+w) 2 =___. We can then take the square root of both sides and solve for x. Note: before completing the square we must make the coefficient of x 2 1. Try to factor: 2*(-7)=-14 (-2)*7= *1=-14 (-1)*14=-14

Completing the Square

Completing the Square Continued

Quadratic Formula

Quadratic Formula – the Discriminant Because of this “discriminating ability,” b 2 - 4ac is called the discriminant. If b2-4ac < 0, there are two conjugate imaginary roots If b2-4ac = 0, there is one real root (called a double root) If b2-4ac < 0, there are two different real roots

Selecting a Method to Solve a Quadratic Equation SituationMethod to use a, b, and c are integers and b 2 - 4ac is a perfect square i.e. is an integer Factoring The equation has the form: x 2 + (even number)x + constant = 0 Completing the Square All other casesQuadratic Formula

Losing and Gaining Roots We must be careful when solving quadratic equations because we can Lose roots (we don’t find all the roots) and we can gain roots Gain roots (we find roots that don’t actually exist)

Losing Roots Roots (solutions) can be lost when dividing both sides of an equation by a common factor Ex) How to prevent this: 1. Bring all terms to one side and solve for zero 2. Solve by cases Notice: x=1 is a solution, for when 1 is input for x, we get 0=0 so we lost the root x=1 when we divided by (x-1)

Losing Roots Bring all terms to one side and solve for zero Solve by cases: If x-1=0 (x=1) If x-1≠0 (x≠1)

Gaining Roots We can gain roots that don’t actually exist by multiplying both sides of an equation by an expression or by squaring both sides of an equation Multiplying both sides of an equation by an expression Ex) Squaring both sides of an equation Ex)

Homework p35: 1, 3, 5, 7, 11, 13, 16, 17,18, 20, 25, 30, 32 Extra Credit: 35 and 44