Solving Quadratic Equations

Solving Quadratic Equations What is the definition of a solution to a quadratic equation? A solution is the value of x when y = 0. What are other terms for the solutions to a quadratic equation? roots zeros x-intercepts (in some cases)

Graphs and Solutions to Quadratic Functions: 3 Cases Two x-intercepts = two real solutions (rational or irrational) One x-intercept = one real solution (always rational) No x-intercepts = two complex solutions ALL QUADRATIC EQUATIONS CAN BE SOLVED!

Name the 4 methods of solving quadratics factoring The Square Root The Quadratic Formula Completing the Square

Method 1: Solving Quadratic Equations by Factoring ok ok need this to be 0 Let's solve the equation 1. First you need to get it in what we call “standard form" which means Factoring is the easiest way to solve a quadratic equations, but it won’t work for all functions, as many cannot be factored! Subtract 18 2. Now let's factor the left hand side 3. Now set each factor = 0 and solve for each answer. Meaning: 2 x-intercepts, 2 real solutions

Method 2: The Square Root: ax2 + c = 0 This method will work for any equation that doesn’t have a “bx” term, it only has “ax2” or “a(x-h)2” and a constant. The objective is to get x2 alone on one side of the equation and then take the square root of each side to cancel out the square. 1. Get the "squared stuff" alone which in this case is the t 2 25 5 5 2. Now square root each side. Don’t forget that (-5)(-5) = 25 also! Meaning: 2 x-intercepts, 2 real solutions

Let's try another one: 1. Get the "squared stuff" alone which in this case is the u 2 4 4 2. Square root each side. Remember with a fraction you can square root the top and square root the bottom DON'T FORGET BOTH THE + AND – Hey, what about the – under the square root? Recall , so x equals two imaginary numbers! Meaning: no x-intercepts, but there are still 2 solutions.

You try

Another Example: “a(x-h)2” 1, Get the "squared stuff" alone (i.e, the parentheses) 2. 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 Meaning: two x-intercepts, but they are irrational.

Perfect Square Trinomials: What’s the pattern? Add how much? c = ? Factored form 1 4 9 16 “add half of b squared” To complete the square and make a perfect square trinomial,_________________

What completes the square? 100 36 81/4 You can only complete the square when a = 1! No Solution

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

“x equals opposite b plus or minus square root of b squared minus 4ac all over 2a”

two complex solutions (no x-intercepts) This part of the formula is called the “Discriminant” The discriminant tells us what kind of solutions we have: One real solution one x -intercept (always rational) two complex solutions (no x-intercepts) Two real solutions two x-intercepts (rational or irrational)

The Quadratic Formula (4) 1. Identify a, b, c 6. Simplify a = 4 b= 2 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

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

Another example Solve the equation

“No solution” could never be an answer to a quadratic equation. True or False? All quadratic equations have solutions. “No solution” could never be an answer to a quadratic equation. TRUE. You can solve ANY quadratic equation, you just may need to use a particular method to get to the answers.

All quadratics have solutions. It is the value of x when y = 0. True or False? The solutions (zero, root) to a quadratic equation are always x-intercepts on its graph. False All quadratics have solutions. It is the value of x when y = 0. But not all quadratics cross the x – axis, so the solutions will not always be x-intercepts.

Neither of these functions have x-intercepts, but they still have two complex solutions

True or False? False All quadratics equations can be factored. Here are just a few examples of quadratics that cannot be factored:

False. You will need to have the formula memorized. True or False? The quadratic formula will be provided to you for the test and final exam. False. You will need to have the formula memorized.

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. 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

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.