Extracting Square Roots © 2002 by Shawna Haider

Solving Quadratic Equations by Factoring need this to be 0 1 ok ok Let's solve the equation -18 -18 First you need to get it in what we call "quadratic form" which means So we have Now let's factor the left hand side Now set each factor = 0 and solve for each answer.

Extracting Square Roots The idea behind this method is when you have some "stuff" squared that you can get by itself on the left hand side of the equation (no other variables on the right hand side), you can then take the square root of each side to cancel out the square. 25 Get the "squared stuff" alone which in this case is the t 2 5 5 Now square root each side. Since you loose any negative sign when you square something, both the + and – of the number would solve the equation so you must do both.

Let's try another one Get the "squared stuff" alone which in this case is the u 2 4 4 Now square root each side and DON'T FORGET BOTH THE + AND – Remember with a fraction you can square root the top and square root the bottom

DON'T FORGET BOTH THE + AND – Another Example Get the "squared stuff" alone which in this case is the stuff in the parenthesis and it is alone. Now square root each side and DON'T FORGET BOTH THE + AND – 25 · 2 Let's simplify the radical -1 -1 Now solve for x 2 2

DON'T FORGET BOTH THE + AND – One Last Example Get the "squared stuff" alone which in this case is the stuff in the parenthesis. -25 -25 Now square root each side and DON'T FORGET BOTH THE + AND – This will give you an i +3 +3 Now solve for y 2 2

Method 4: The Quadratic Formula The Quadratic Formula is a formula that can solve any quadratic, but it is best used for equations that cannot be factored or when completing the square requires the use of fractions. It is the most complicated method of the four methods. Do you want to see where the formula comes from?

The Quadratic Formula 5. Simplify radical 1. Divide by a This formula comes from completing the square of a quadratic written in standard form 5. Simplify radical 1. Divide by a 6. Get x alone 2. Subtract c/a and add half of b/a squared 7. Simplify right hand side 3. Factor left side, combine right side 4. Square root each side

The Quadratic Formula a = 4 b= -2 c = 5 (4) 1. Identify a, b, c Solve the equation Notice the solutions are complex! 1. Identify a, b, c 6. Simplify a = 4 b= -2 c = 5 =4•19 2. Plug into the formula 7. Simplify radical 2 (2)2 (4) (5) (4) 8. Simplify final answer, if possible 5. Simplify Meaning: 0 x-intercepts, 2 complex solutions

The Quadratic Formula ( ) 1. Identify a, b, c 6. Simplify a = b= c = Solve the equation 1. Identify a, b, c 6. Simplify a = b= c = 2. Plug into the formula 7. Simplify radical ( )2 ( ) ( ) ( ) 8. Simplify final answer, if possible 5. Simplify

Another example Solve the equation

Completing the Square Isolate the terms with variables on one side of the equation, and arrange them in descending order. Divide both sides by the coefficient of x² if that coefficient is not 1. Complete the square by taking half of the coefficient of x and adding its square to both sides.

Express the trinomial as the square of a binomial (factor the trinomial) and simplify the other side. Use the principle of square roots (find the square roots of both sides). Solve for x by adding or subtracting on both sides.

Example Only variable terms are on the left side. Subtracting 4 to both sides. We can now complete the square on the left side. Completing the square: ½(6) = 3 and (3)² = 9 Factoring and simplifying Using the principle of square roots.

Completing the Square: Use #1 This method is used for quadratics that do not factor, although it can be used to solve any kind of quadratic function. 1. Get the x2 and x term on one side and the constant term on the other side of the equation. 2. To “complete the square,”, add “half of b squared” to each side. You will make a perfect square trinomial when you do this. 3. Factor the trinomial 4. Apply the square root and solve for x

Completing the Square: Use #2 By completing the square, we can take any equation in standard form and find its equation in vertex form: y = a(x-h)2 + k 1. Get the x2 and x term on one side and the constant term on the other side of the equation. 2. To “complete the square,”, add “half of b squared” to each side. You will make a perfect square trinomial when you do this. What is the vertex? 3. Factor the trinomial. 4. Write in standard form.

Which method should you use? Solve (x+1)(x+5) b. x = -1, x = -5 c. x = 1, x = 5 d. no sol Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem? The Square Root factoring The Quadratic Formula

Which method should you use? Solve (x+1)(x+5) b. x = -1, x = -5 c. x = 1, x = 5 d. no sol Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem? The Square Root factoring The Quadratic Formula

Which method should you use? Solve b. c. no sol because you cannot factor it Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem? The Square Root factoring The Quadratic Formula

Which method should you use? Solve b. c. d. Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem? The Square Root factoring The Quadratic Formula

Which method should you use? a. b. c. Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem? The Square Root factoring The Quadratic Formula

Applications of Quadratic Equations © 2002 by Shawna Haider

500 500 COMPOUND INTEREST P = $500 and A = $572.45 Amount in account after two years Interest rate as a decimal Principal Amount you deposit P = $500 and A = $572.45 Let's substitute the values we are given for P and A Solve this equation for r 500 500 Square root both sides but don't need negative because interest rate won't be negative

PYTHAGOREAN THEOREM An L-shaped sidewalk from building A to building B on a college campus is 200 feet long. By cutting diagonally across the grass, students shorten the walking distance to 150 feet. What are the lengths of the two legs of the sidewalk? 200-x Draw a picture: B x If first part of sidewalk is x and total is 200 then second part is 200 - x 150 A Using the theorem: Multiply out continued on next slide

use the quadratic formula to solve get everything on one side = 0 divide all terms by 2 use the quadratic formula to solve 1 200 - 134.5 = 64.6 so doesn't matter which you choose, the two lengths are 135.4 meters and 64.6 meters.

After how many seconds will the height be 11 feet? Height of a tennis ball A tennis ball is tossed vertically upward from a height of 5 feet according to the height equation where h is the height of the tennis ball in feet and t is the time in seconds. After how many seconds will the height be 11 feet? So there are two answers: (use a calculator to find them making sure to put parenthesis around the numerator) t = .42 seconds or .89 seconds. Get everything on one side = 0 and factor or quadratic formula. -11 -11

When will the tennis ball hit the ground? What will the height be when it is on the ground? h = 0 So there are two answers: (use a calculator to find them) t = - 0.21 or 1.52 seconds (throw out the negative one)

Average Speed A truck traveled the first 100 miles of a trip at one speed and the last 135 miles at an average speed of 5 miles per hour less. If the entire trip took 5 hours, what was the average speed for the first part of the trip? If you used t hours for the first part of the trip, then the total 5 minus the t would be the time left for the second part. Let's make a table with the information first part second part distance rate time r 100 t 135 r - 5 5 - t

Use this formula to get an equation for each part of trip Distance = rate x time distance rate time r first part 100 t second part 135 r - 5 5 - t Solve first equation for t and substitute in second equation 100 = r t 135 = (r - 5)(5 - t) r r

r r r r r FOIL the right hand side Multiply all terms by r to get rid of fractions r r r r Combine like terms and get everything on one side Divide everything by 5 Factor or quadratic formula So r = 50 mph since r = 2 wouldn't work for second part where rate is r –5 and that would be –3 if r was 2.