By Sammi Smith and Hanna Van Pelt Quadratic Equations Graphing Standard Form Standard form is ax^2+bx+c=y. A shows the width of the parabola, b shifts the parabola right or left, and c shows the y-intercept. Solving Quadratic Equations by Factoring Factoring a quadratic equation will get you your roots. To factor, your roots will end up to be factors of c, and when added together they will equal the coefficient of b. The roots, or 'zeros', are the points where the parabola crosses the x-axis. Solving Quadratic Equations with Square Root Methods Explanation: You use the square roots method when there is NO b value. A root or 'zero' are the points where the parabola crosses the x- axis. Finding the Axis of Symmetry The axis of symmetry is the line that runs vertically, directly through the center of the parabola. The formula to find the axis of symmetry is: . x=-b/2a Finding the Vertex As we said before, to find the axis of symmetry you use the formula -b/2a=x. That gives you the x value of the coordinate. To find the y value you must plug the x value back into the quadratic equation. Example: you were given the quadratic equation of x^2+4x+2=y. you would then use the axis of symmetry formula: -4/2(1)= 2. Then you would plug 2 back into your original equation: 2^2+4(2)+2=14. 14 is your y value in the vertex and 2 is your x value. The coordinates of the vertex are (2,14). Finding the Roots To find the roots, (x-intercepts), you would look on the graph. The points where the parabola crosses the x-axis are your roots. Examples of a Table and Graph Equation: x^2-6x+5 To graph, all you need to do is plug in values for x and get the y values. You then connect the points to get your parabola. How to Solve by using Square Roots As stated before, you only use square root method when there is no b value in your quadratic equation. Since a is always squared you would subtract the c value from the left side of the equation, as well as the right. Then you would divide the a coefficient so that on the left side there is only x^2 and on the right side there is only the c value. You then find the square root of c, to get the value of x. Example of Solving Equation: 4x^2-36=0

Step one, move the c value to the other side of the equation: 4x^2=36

Step 2, divide c by the coefficient of a: x^2=9

Step 3, find the square root of c, which will get you the value of x: x=3 Examples of Solving when you get a Radical Equation: 3x^2-36=0

Step 1, move c to the other side of the equation: 3x^2=36

Step 2, divide by the coefficient of a: x^2=12

Step 3, find the square root of c:x=+/- 12

If c is not a perfect square, then just simplify the radical, and have your final answer be plus OR minus the value of the radical. The final answer after you simplify is x=+/-2 3 Equation: 2x^2-98=0

2x^2=98

x^2=49

x=+/-7 Methods of Factoring Grouping Factoring by grouping is a good option when you have 4 or more terms. Example: Greatest Common

Factor Khan Academy Factoring by Grouping GCF or greatest common factor is a number that goes into all terms. You put the GCF outside of parenthesis and distribute it to all of the terms. GCF CAN be x. Example: 12x^2+15x+3

Step one, put GCF outside of parenthesis (GCF is 3): 3(4x^2+5x+1)

When you have completed factoring out the GCF you should be able to distribute and get your original equation Trial and Error This is probably the most common way to factor. It is called trial and error because you test out different numbers until you get the right combination. Remember that the roots, or answers, have to be factors of c, and added together to equal the coefficient of b. Example:2x^2-6x-16

Step one, try out some numbers: -4 and 4...?

-4x4=-16 -4+4=0 So -4 and 4 don't work, so try some more. -8 and 2...? -8x2=-16 -8+2=-6 Factoring Shortcuts (Special Cases) Difference of Squares There a a few shortcuts that can be used in factoring: When you have a^2-b^2, you can find the square roots of a and b and then factor it so that there is one positive and one negative per side. Like this: a^2-b^2=(a-b)(a+b) Perfect Squares Solving Quadratic Equations using the Quadratic Formula The roots or 'zeros' are the points where the parabola crosses the x-axis. The Quadratic Formula is: x= -b+/- b^2-4ac 2a Examples of Solving Example 1: Equation: x^2+20x+96 x= -20+/- 20^2-4(96)(1) 2(1) x= -20+/- 16 2 x= -20+/-4 2 x= -12, x=-8 Example of Solving Solving Quadratic Equations by Completing the Square "Move the loose number over to the other side. Divide through by whatever is multiplied on the squared term. Take half of the coefficient (don't forget the sign!) of the x-term, and square it. Add this square to both sides of the equation. Convert the left-hand side to squared form, and simplify the right-hand side. (This is where you use that sign that you kept track of earlier. You plug it into the middle of the parenthetical part.) Square-root both sides, remembering the "±" on the right-hand side. Simplify as necessary. Solve for x."

Explanation from www.purplemath.com Example x^2+6x=16

(6/2)^2=9

x^2+9x=16+9

(x+2)^2=25

(x+3)^2 = 25 x+3=+/-5

x=-8,2 Example x^2-8x-20

x^2-8x=20

(-8/2)^2=16

x^2-8x+16=20+16

(x+4)^2= 36

x+4=+/-6

x=2,-10 Example x^2+30x-75

x^2+30x=75

x^2+30x+225=175+225

(x+15)^2= 400

x+15=+/-20

x=-35,5 The roots, or 'zeros', are the points where the parabola crosses the x-axis What are roots? 'Real-Life' Problem A ball is thrown into the air from 5 feet, it is thrown up at a velocity of 15 feet/second. When will the ball hit the ground? Hint (use the falling object model):

Height=-16t^2+vt+s Solving Step 1: set up your equation: 0=-16t^2+15t+5

Step 2: solve like a regular quadratic equation: 0=-15+/- 225+320 = O=-15+/- 545

0=-15+/-23

Step 3: find your answers.

x=-38 x=8 It is impossible for x to be negative, because we are dealing with a real life situation. These final answers are approximate. You never want to leave an answer with a radical when solving a word problem. What do these answers mean? Before you give answers, you must remember what you are solving for. In this case, it is x, or time. When you write the answers, you have to label it with seconds. Also note... Graph Example of Solving

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# Quadratic Equations

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