PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS QUADRATIC – means second power Recall LINEAR – means first power

Click on the Quadratic Method you wish to review to go to those slides Quadratic Formula Quadratic Formula – Always works, but be sure to rewrite in standard form first of ax 2 + bx + c = 0 How to Choose Among Methods 1 – 4Methods 1 – 4? Summary & Hints Quadratic Functions Quadratic Functions – Summary of graphing parabolas from f(x) = ax 2 + bx + c form Factoring Method Factoring Method – Quick, but only works for some quadratic problems Square Roots of Both Sides Square Roots of Both Sides – Easy, but only works when you can get in the form of (glob )2 = constant Complete the Square Complete the Square – Always works, but is recommended only when a = 1 or all terms are evenly divisible to set a to 1.

METHOD 1 - FACTORING Set equal to zero Factor Use the Zero Product Property to solve (Each factor with a variable in it could be equal to zero.)

METHOD 1 - FACTORING Any # of terms – Look for GCF factoring first! 1. 5x 2 = 15x 5x 2 – 15x = 0 5x (x – 3) = 0 5x = 0 OR x – 3 = 0 x = 0 OR x = 3 {0, 3}

METHOD 1 - FACTORING Binomials – Look for Difference of Squares 2. x 2 = 9 x 2 – 9 = 0 (x + 3) (x – 3) = 0 x + 3 = 0 OR x – 3 = 0 x = – 3 OR x = 3 {– 3, 3} Conjugates

METHOD 1 - FACTORING Trinomials – Look for PST (Perfect Square Trinomial) 3. x 2 – 8x = – 16 x 2 – 8x + 16 = 0 (x – 4) (x – 4) = 0 x – 4 = 0 OR x – 4 = 0 x = 4 OR x = 4 {4 d.r.} Double Root

METHOD 1 - FACTORING Trinomials – Look for Reverse of Foil 4. 2x 3 – 15x = 7x 2 2x 3 – 7x 2 – 15x = 0 (x) (2x 2 – 7x – 15) = 0 (x) (2x + 3)(x – 5) = 0 x = 0 OR 2x + 3 = 0 OR x – 5 = 0 {-3/2, 0, 5} x = 0 OR x = – 3/2 OR x = 5

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METHOD 2 – SQUARE ROOTS OF BOTH SIDES Reorder terms IF needed Works whenever form is (glob) 2 = c Take square roots of both sides (Remember you will need a  sign!) Simplify the square root if needed Solve for x. (Isolate it.)

METHOD 2 – SQUARE ROOTS OF BOTH SIDES 1. x 2 = 9 x =  3 Note  means both +3 and -3! x = -3 OR x = 3 {-3, 3}

METHOD 2 – SQUARE ROOTS OF BOTH SIDES 2. x 2 = 18

METHOD 2 – SQUARE ROOTS OF BOTH SIDES 3. x 2 = – 9 Cannot take a square root of a negative. There are NO real number solutions!

METHOD 2 – SQUARE ROOTS OF BOTH SIDES 4. (x-2) 2 = 9 This means: x = and x = 2 – 3 x = 5 and x = – 1 {-1, 5}

METHOD 2 – SQUARE ROOTS OF BOTH SIDES Rewrite as (glob) 2 = c first if necessary. 5. x 2 – 10x + 25 = 9 x = 8 and x = 2 {2, 8} (x – 5) 2 = 9

METHOD 2 – SQUARE ROOTS OF BOTH SIDES Rewrite as (glob) 2 = c first if necessary. 6. x 2 – 10x + 25 = 48 (x – 5) 2 = 48

PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES 1. x 2 = 121 x =  11 Note  means both +11 and -11! x = -11 OR x = 11 {-11, 11}

2. x 2 = – 81 Square root of a negative, so there are NO real number solutions! PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES

Rewrite as (glob) 2 = c first if necessary. 3. 6x 2 = 156 x 2 = 26

4. (a – 7) 2 = 3 PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES

Rewrite as (glob) 2 = c first if necessary. 5. 9(x 2 – 14x + 49) = 4 {6⅓, 7⅔} (x – 7) 2 = 4/9

PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES 6.

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METHOD 3 – COMPLETE THE SQUARE Goal is to get into the format: (glob) 2 = c Method always works, but is only recommended when a = 1 or all the coefficients are divisible by a We will practice this method repeatedly and then it will keep getting easier!

METHOD 3 – COMPLETE THE SQUARE Example: 3x 2 – 6 = x x 2x 2 – 12x – 6 = 0 Simplify and write in standard form: ax 2 + bx + c = 0 x 2 – 6x – 3 = 0Set a = 1 by division Note: in some problems a will already be equal to 1.

METHOD 3 – COMPLETE THE SQUARE (x – 3) 2 = 12Rewrite as (glob) 2 = c x 2 – 6x – 3 = 0 Move constant to other side Leave space to replace it! x 2 – 6x = 3 x 2 – 6x + 9 = 3 + 9Add (b/2) 2 to both sides This completes a PST!

METHOD 3 – COMPLETE THE SQUARE Solve for x Take square roots of both sides – don’t forget  Simplify (x – 3) 2 = 12

PRACTICE METHOD 3 – COMPLETE THE SQUARE Example: 2b 2 = 16b + 6 2b 2 – 16b – 6 = 0 Simplify and write in standard form: ax 2 + bx + c = 0 b 2 – 8b – 3 = 0 Set a = 1 by division Note: in some problems a will already be equal to 1.

PRACTICE METHOD 3 – COMPLETE THE SQUARE (b – 4) 2 = 19Rewrite as (glob) 2 = c b 2 – 8b – 3 = 0 Move constant to other side Leave space to replace it! b 2 – 8b = 3 b 2 – 8b + 16 = 3 +16Add (b/2) 2 to both sides This completes a PST!

PRACTICE METHOD 3 – COMPLETE THE SQUARE Solve for the variable Take square roots of both sides – don’t forget  Simplify (b – 4) 2 = 19

PRACTICE METHOD 3 – COMPLETE THE SQUARE Example: 3n n + 1 = n - 2 3n n + 3 = 0 Simplify and write in standard form: ax 2 + bx + c = 0 n 2 + 6n + 1 = 0 Set a = 1 by division Note: in some problems a will already be equal to 1.

PRACTICE METHOD 3 – COMPLETE THE SQUARE (n + 3) 2 = 8 Rewrite as (glob) 2 = c n 2 + 6n + 1 = 0 Move constant to other side Leave space to replace it! n 2 + 6n = -1 n 2 + 6n + 9 = Add (b/2) 2 to both sides This completes a PST!

PRACTICE METHOD 3 – COMPLETE THE SQUARE Solve for the variable Take square roots of both sides – don’t forget  Simplify (n + 3) 2 = 8

PRACTICE METHOD 3 – COMPLETE THE SQUARE 1. x 2 – 10x = -3 What number “completes each square”? 1. x 2 – 10x + 25 = x x = 12. x x + 49 = x 2 – 1x = 53. x 2 – 1x + ¼ = 5 + ¼ 4. 2x 2 – 40x = 44. x 2 – 20x =

PRACTICE METHOD 3 – COMPLETE THE SQUARE 1. x 2 – 10x + 25 = Now rewrite as (glob) 2 = c 1. (x – 5) 2 = x x + 49 = (x + 7) 2 = x 2 – 1x + ¼ = 5 + ¼3. (x – ½ ) 2 = 5 ¼ 4. x 2 – 20x = (x – 10) 2 = 102

PRACTICE METHOD 3 – COMPLETE THE SQUARE ⅓k 2 = 4k - ⅔ Show all steps to solve. k 2 = 12k - 2 k k = - 2 k k + 36 = (k - 6) 2 = 34

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METHOD 4 – QUADRATIC FORMULA This is a formula you will need to memorize! Works to solve all quadratic equations Rewrite in standard form in order to identify the values of a, b and c. Plug a, b & c into the formula and simplify! QUADRATIC FORMULA:

METHOD 4 – QUADRATIC FORMULA Use to solve: 3x 2 – 6 = x x Standard Form: 2x 2 – 12x – 6 = 0

METHOD 4 – QUADRATIC FORMULA

PRACTICE METHOD 4 – QUADRATIC FORMULA 2x 2 = x + 6 Show all steps to solve & simplify. 2x 2 – x – 6 =0

PRACTICE METHOD 4 – QUADRATIC FORMULA x 2 + x + 5 = 0 Show all steps to solve & simplify.

PRACTICE METHOD 4 – QUADRATIC FORMULA x 2 +2x - 4 = 0 Show all steps to solve & simplify.

THE DISCRIMINANT – MAKING PREDICTIONS Four cases: b 2 – 4ac is called the discriminant 1. b 2 – 4ac positive non-square  two irrational roots 2. b 2 – 4ac positive square  two rational roots 3. b 2 – 4ac zero  one rational double root 4. b 2 – 4ac negative  no real roots

THE DISCRIMINANT – MAKING PREDICTIONS Use the discriminant to predict how many “roots” each equation will have. 1. x 2 – 7x – 2 = = 2x 2 – 3x = 5x 2 – 2x x 2 – 10x + 25=0 49–4(1)(-2)=57  2 irrational roots 9–4(2)(1)=1  2 rational roots 4–4(5)(3)=-56  no real roots 100–4(1)(25)=0  1 rational double root

THE DISCRIMINANT – MAKING PREDICTIONS about Parabolas The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have and where the vertex is located. 1. y = 2x 2 – x f(x) = 2x 2 – x y = -2x 2 – 9x f(x) = x 2 – 6x + 9 1–4(2)(-6)=49  2 rational zeros opens up/vertex below x-axis/2 x-intercepts 1–4(2)(6)=-47  no real zeros opens up/vertex above x-axis/No x-intercepts 81–4(-2)(6)=129  2 irrational zeros opens down/vertex above x-axis/2 x-intercepts 36–4(1)(9)=0  one rational zero opens up/vertex ON the x-axis/1 x-intercept

THE DISCRIMINANT – MAKING PREDICTIONS Note the proper terminology: The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have. The “roots” of an equation are the x values that make the expression equal to zero. Equations have roots. Functions have zeros which are the x-intercepts on it’s graph.

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FOUR METHODS – HOW DO I CHOOSE? Some suggestions: Quadratic Formula – works for all quadratic equations, but look first for a “quicker” method. Don’t forget to simplify square roots and use value of discriminant to predict number of roots. Square Roots of Both Sides – use when the problem is easily written as glob 2 = constant. Examples: 3(x + 2) 2 =12 or x 2 – 75 = 0

FOUR METHODS – HOW DO I CHOOSE? Some suggestions: Factoring – doesn’t always work, but IF you see the factors, this is probably the quickest method. Examples:x 2 – 8x = 0 has a GCF 4x 2 – 12x + 9 = 0 is a PST x 2 – x – 6 = 0 is easy to FOIL Complete the Square – best used when a = 1 and b is even (so you won’t need to use fractions). Examples: x 2 – 6x + 1 = 0

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a.The graph is a parabola. Opens up if a > 0 and down if a < 0. b.To find x-intercepts: – may have Zero, One or Two x-intercepts 1. Set y or "f(x)" to zero on one side of the equation 2. Factor & use the Zero Product Prop to find TWO x-intercepts c.To find y-intercept, set x = 0. Note f(0) will equal c. I.E. (0, c) d.To find the coordinates of the vertex (turning pt): 1. x-coordinate of the vertex comes from this formula: 2. plug that x-value into the function to find the y-coordinate e. The axis of symmetry is the vertical line through vertex: x = REVIEW – QUADRATIC FUNCTIONS

Example Problem: f(x) = x 2 – 2x – 8 a. Opens UP since a = 1 (that is, positive) b. x-intercepts:0 = x 2 – 2x – 8 0 = (x – 4)(x + 2) (4, 0) and (– 2, 0) c. y-intercept: f(0) = (0)2 – 2(0) – 8  (0, – 8) d. vertex: e. axis of symmetry: x = 1

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