Christa Walters 9-5

How to factor a polynomial? A polynomial is an algebraic expression that consists of 2 or more terms. Steps: Find the GCF Factor it in every number Remember that it is like reverse FOILing.

examples −3x² + 15x = −3x ( x − 5 ) 2x²-4x+6 = 2(x²-2x+3)

What’s a quadratic function? f(x) = ax2 + bx + c In the form of : A,B, and C are all numbers, and A can never equal to 0 The graph of a quadratic function is a parabola ( a curve) which will vary on steepness, width and being upward or downward. The quadratic equation is in standardize form, ax²+bx+c=0, and a will never = 0 because it if equals the x² disappears and it would be a linear function. If it is a function it would pass the vertical line test. The difference between a quadratic function and a linear function is that a quadratic is a curve (parabola) and the linear is a straight line. Quadratic = ax²+bx+c=0 Linear = y=mx+b

Examples 1. y = -x² 2. y = 3x² + 6x + 1 3. y = 0.5x²

How to graph a quadratic function? To graph a parabola, in order to solve a quadratic function, you need to know what formula to use. FORMULA: a(x+b)² +c=0 A- Changes the steepness. If a is - the parabola will be going down, if a is + it will be going up. ( a<1 – skinnier parabola. a>1 – bigger parabola. B- moves right and left. the b units are plotted opposite from its sign. Ex. -2 would be plotted on 2. C- moves the vertex up or down (c units= +c -c )

Examples: 1. y = -2(x-4)² + 5 2. y = (x-3)² +4 3. y = 2(x -4)² +1

How to solve a quadratic What’s solution? The answer to an equation. by graphing it? When you have your equation ready, write down what # a,b and c are -b/2(a) : you just divide negative b and then divide it by ax2 when you get the answer to that create a T-table the first x value will be the answer you got from what = -b/2(a) Plug that in to the x values and what you get from that will be the y value. write down more x values and do the same Lastly plot the points and connect the dots to create a parabola. ng What’s solution? The answer to an equation. A parabola will have no solution if it doesn’t touch the x-intercept.

Examples 2. y=x²+2x+5 3. y=-x²-8x-17 1. y = x² + 3x + 2 -b/2a= -3/2=-1.5 X Y -1.5 -2.5 -1 0 0 2 -b/2a= -2/2=-1 X Y -1 4 0 5 1 8 2 13 -b/2a= -4 X Y -4 -1 -3 -2 -2 -5 No solution No solution X= -1,-2

Equations using square roots? How to solve quadratic Equations using square roots? STEPS: Get x² by itself Make sure there is no X square root both sides DO NOT FORGET THE +/- Ex. √x² = √64 X= 8, -8 x=+/- 8

examples x2 – 4 = 0 2. -4x²+64 =0 x2 = 4 -4x²=64 √ x² √4 √x²=√16

Equations using factoring? How to solve quadratic Equations using factoring? FORMULA: ax²+bx+c To make it easier for you first, write what numbers are A,B, and C. Then multiply A and C. Find two numbers that multiply to what you got. Those two numbers also have to add up to B. When you have those two numbers, put them both over A and then reduce. Write your answer as: ( #x +#) (#x+#)

examples a= 3 b=1 c=-24 a= 3 b= -2 c=-16 3x²+x-24 a= 2 b= 11 c= 12 -72 1 9 , -8 put both over a (3) 3 3 3 reduce 1 (1x+3)(3x-8) X= -3,8 a= 3 b= -2 c=-16 3. 3x²-2x-16 -48 -2 -8 , 6 3 3 2 1 (1x+2)(3x-8) X= 8 or -2 a= 2 b= 11 c= 12 2. 2x²+11x+12 24 11 8 , 3 2 2 4 1 (1x+4)(2x+3) X= -4,-3

How to solve a quadratic equation using Completing the ? STEPS TO SUCCED: Get x=1 Get c by itself Complete the square ( get a=1, find b, divide by 2, square it, (b/2)², factor (x+ b/2) ² ) Add (b/2) ² to both sides √ both sides ,and don’t forget +/-

examples 2. a ² -2a-8=0 +8 +8 a ²-2a = 8 +1 √(a-1)= √9 a-1 = +/-3 +8 +8 a ²-2a = 8 +1 √(a-1)= √9 a-1 = +/-3 +1 +1 a= 4,-2 3. m ² -12m +26=0 -26 -26 m ²-12m=26 +36 √(m-6) ² = √10 m-6 = √10 +6 +6 m= 6 +/- √10 x ² -10x + 16 = 0 -16 -16 x ²-10x=-16 +25 √(x -5) ² = √9 X-5= +/- 3 +5 +5 x= 8, 2

Quadratic formula It ‘s=(b2 - 4ac) Using the quadratic formula, is very easy, you just have to plug in the numbers where the letters are and SOLVE!!!! Discriminant of an equation: It gives you an idea of how many square roots an equation has. It will tell you how many x-intercepts the quadratic equation will have. It ‘s=(b2 - 4ac) 

Examples 1. x ² -8x+7=0 x= 8 +/- √(-8 ²) -4(1)7 x= 8+/-√64-28 x=8+/- √36 x= 8+/- 6 x=7,1 2(1) 2 2 2 4+/-3= 2. x ²-8x+12=0 x= 8 +/- √(-8 ²) -4(1)12 x= 8+/-√64-48 x=8+/- √16 x= 8+/- 4 x=6,2 2(1) 2 2 2 4+/- 2= 4x ² -8x +4=0 x= 8 +/- √(-8 ²) -4(4)4 x= 8+/-√64-64 x=8+/- √0 x= 8+/- 0 x=1 2(4) 8 8 8 1+0=1