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Quadratics Solve quadratic equations using multiple methods: factoring, graphing, quadratic formula, or square root principle.

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What is a Quadratic?? The standard form for any quadratic equation is ax2 + bx + c = 0 There are many ways to solve quadratic equations. Below are the ways that we will solve them. Factoring Graphing Quadratic Formula Square root principle

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First, FACTORING!! We will first factor a quadratic expression and solve for the unknown in the equation. We will begin to factor quadratic expressions with a = 1.

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**Put your answers in set notation**

First problem! Always goes with the largest number!! means same signs! x x = 0 ( x ) ( x ) =0 Now set each set of parentheses equal to zero. x + 3 = x + 4 = 0 x = x = -4 Put your answers in set notation { -3 , -4} + + 3 4

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**Now let’s try this one! - x x x2 + 8x - 20 = 0 ( )( ) =0 + 2 10**

Always goes with the largest number Means different signs x x = 0 ( )( ) =0 - x + x 10 2

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**Second problem with a ≠ 0. 5x2 + 27x +10 = 0 5x2 + 25x + 2x +10 =0**

ALWAYS GOES WITH THE LARGER NUMBER!! SAME SIGNS!!!!! 5x2 + 27x +10 = 0 5x2 + 25x + 2x +10 =0 (5x2+25x)+(2x+10)=0 5x(x + 5) + 2(x + 5) =0 (x + 5) (5x + 2) = 0 x + 5 = x + 2 = 0 x = x = -2/5 {-5,-2/5} First, multiply a and c together. 5 * 10 = 50 Second, Ask yourself what are the factors of 50 that will add or subtract to give you b? – Let’s list them 1*50 = 50 2*25 = 50 5*10 = 50 Which set of factors can add to give you 27? Correct! 2 and 25 Therefore we will have +25 and +2 Now, group so it will be easy to factor! Since there are two (x+5), write them one time! And also write the GCF of each in a set of parenthesis by themselves! LAST, set each factor equal to zero and solve!

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**Try this one! 24x2 – 22x + 3 = 0 ANSWER: (4x – 3)(6x – 1) = 0**

{3/4,1/6}

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GRAPHING Another way to solve a quadratic equation is to graph it! To graph a quadratic equation, you must have a domain. You can pick your own domain or it will be given!

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y = 2x2 – 4x – 5 X Y -2 -1 1 2 3 4 11 1 -5 -7 -5 1 11 vertex

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**Finding the solutions after graphing!**

To find the solutions of the quadratic equation on a graph, look where the parabola intersects the x-axis!

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**Here is an example using the calculator!**

y = x2 – x – 2 So, the solutions are -1 and 2 {-1, 2}

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**Or just look at the table!!**

The solutions or ROOTS can be found in the table by finding where the y value is zero! And the ROOTS are {-1, 2}

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Looking at graphs! You can look at graphs and tell how many roots they have! Here’s how!! This graph has no solution because the parabola NEVER crosses the x-axis

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**This graph touches the x-axis one time**

This graph touches the x-axis one time. Therefore, we say that it has a double root!!

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The Quadratic Formula!!! Yet another way to solve a quadratic equation is to use the QUADRATIC FORMULA!!!!!!

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The Quadratic Formula

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**Using the Quadratic Formula**

X2 – 2x – 24 =0 A= 1 B= C= -24 Now plug in the numbers into the formula! Now just plug this into the calculator!

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Calculator steps Notice the subtraction sign Notice the addition sign

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**The Square Root Principle**

Another way to solve a quadratic equation is to use the square root principle! Some equations can be solved by taking the square root of both sides!

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**Using the square root principle!**

x2 – 10x + 25 = 7 ( )( )=7 (x – 5)2 = 7 Steps: Factor the left side. Write the factors one time Now take the square root of both sides. Now solve the two problems! (Use your calculator) x - x - 5 5

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Calculator steps x= 2.4 x= 7.6 {2.4,7.6}

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**Pick 3 ways out of the 4 to solve this problem!**

x2 – 5x – 24 = 0 The answer you should get if you work the problem three ways is {-3,8}

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