Quadratics     3102.3.30  Solve quadratic equations using multiple methods: factoring, graphing, quadratic formula, or square root principle.

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

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.

Put your answers in set notation First problem! Always goes with the largest number!! means same signs! x2 + 7x + 12 = 0 ( x ) ( x ) =0 Now set each set of parentheses equal to zero. x + 3 = 0 x + 4 = 0 x = -3 x = -4 Put your answers in set notation { -3 , -4} + + 3 4

Now let’s try this one! - x x x2 + 8x - 20 = 0 ( )( ) =0 + 2 10 Always goes with the largest number Means different signs x2 + 8x - 20 = 0 ( )( ) =0 - x + x 10 2

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 = 0 5x + 2 = 0 x = -5 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!

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

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!

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

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

Here is an example using the calculator! y = x2 – x – 2 So, the solutions are -1 and 2 {-1, 2}

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}

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

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!!

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

The Quadratic Formula

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

Calculator steps Notice the subtraction sign Notice the addition sign

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!

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

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

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}