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Complete The Square (x+4)(x-3)=0 FOIL X2 – 5x +4 CRASH COURSE IN QUADRATICS In preparation for the Algebra CST Multiplying Polynomials Area Model of Multiplication To multiply 68 x 34: •Write the two numbers in expanded notation and multiply one box at a time. •After you have multiplied the numbers, add all of the products together. 60 30 + 4 + 8 (30)(60) 1800 (30)(8) 240 (4)(60) 240 (4)(8) 32 1800+240+240+32=2312 Now you try one… 48 x 53 Multiplying Polynomials Area Model of Multiplication To multiply (x+2)(x+3): •Write the two numbers in expanded notation and multiply one box at a time. •After you have multiplied the numbers, add all of the products together. x x + 3 + 2 (x)(x) x2 (x)(2) 2x (3)(x) 3x (3)(2) 6 X2 + 5x + 6 Now you try one… (x+5)(x+1) Multiplying Polynomials FOIL ( x + 2 ) ( x + 3) First (x)(x) = x2 Outer (x)(3) = 3x Inner (2)(x) = 2x Last (2)(3) = 6 Combine like terms… = x2 + 5x + 6 Multiplying Polynomials x2 + 5x + 6 a=1 b=5 c=6 ax2 + bx + c Factoring Polynomials 10 12 3 4 2 5 7 7 3 6 5 6 6 14 1 7 2 7 9 18 21 9 10 4 Ask yourself… “What two numbers multiplied together give you the top digit and added together give you the bottom?” Factoring Polynomials X2 + 7x + 12 X2 + 13x + 36 X2 - 6x - 40 12 7 36 13 -40 -6 (x + )(x+ ) (x + )(x+ ) (x + )(x+ ) Perfect Square Trinomial X2 + 12 + 36 X*X (x + 6)2 6 * 6 X2 -- 14 + 49 X*X (x + 6)(x + 6) 7 * 7 (x - 7)(x - 7) (x - 7)2 Solving Quadratic Equations •Graphing •Factoring •Using Square Roots •Completing the Square •Quadratic Formula Graphing Quadratic Equations x2 – 4x = 0 x y=x2 - 4x y x, y 0 02 – 4(0) 0 0, 0 2 22 - 4(2) -4 2, -4 4 42 – 4(4) 0 4, 0 The Solution is the ________________ Find the solution for each graph: Factoring Quadratic Equations Using the Zero Product Property (x-3)(x+7)=0 (x-3)=0 x=3 (x+7)=0 x = -7 Factoring Quadratic Equations Solve using the Zero Product Property (x-3)(x+4)=0 Can you solve in your head? (x-2)(x+1)=0 (x+3)(2x-8)=0 (3x-1)(4x+1)=0 (3x+1)(8x-2)=0 x2 + 12x + 36 x2 - 21x = 72 -72 -21 x= x= If x2 is added to x, the sum is 42. What are the values of x? Using Square Roots Square-Root Property x2 = 16 4x2 – 25 = 0 +25 +25 √x2 = √16 √4x2 = √25 x = +4 2x = 5 2 2 x2 = 16 (4)2= 16 (-4)2 = 16 x = + 2.5 4x2 = 25 Completing the Square Using Algebra Tiles x2 + 6x a= 1 b=6 b 2 ( ) x2 + 6x = 0 +9 +9 2 b=6 c=0 x2 + 6x + 9 = 9 (x+3)(x+3)=9 (x+3)2 = 9 √(x+3)2 = √ 9 x+3 = + 3 2 ( ) ( 62 ) 6 2 x+3= 3 x=0 x+3= -3 x = -6 2 =9 Completing the Square Add to both sides of the equation x2 Factor the Perfect Square + 14x = 15 + 49 + 49 2 (142) b = 14 = 72 =49 x2 + 14x + 49 = 64 (x+7)(x+7)=64 (x+7)2 = 64 √(x+7)2 = √ 64 x+7 = + 8 x+7= 8 x=1 x+7= -8 x = -15 Completing the Square 3x2 – 10x = -3 3 3 3 Reduce x2 - 10x = -1 3 Factor the Perfect Square x2 - 10x = -1 3 +25 +25 9 9 x2 - 10x + 25 = 16 3 2 9 9 x–5 16 = 3 9 ( b = 10 3 √( x–5 3 ) =√169 3 2 2 = 100 36 25 9 Add to both sides of the equation -9 + 25 = 16 9 9 9 ) 2 (-10 * 1) (x – 53 )=+ 43 x–5=4 3 3 x=9 3 x – 5 = -4 3 3 x=1 3 Completing the Square x 2 - 8x = 12 x 2 - 8x = 5 2 x + 4x = 6 2 x - 4x = 8 2 ax – bx = c What should be added to both sides of this equation? The Quadratic Formula x2 + 5x + 6 2x2 + 3x – 5 = 0 ax2 + bx + c x= -b + √ b2 – 4ac 2a x= -b + √ b2 – 4ac 2a a=1 b=5 c=6 ax2 + bx + c a = 2 b = 3 c = -5 x= -3 + √ 32 – 4(2)(-5) 2(2) x= -3 + √ 9 – (-40) x= -3 + √ 49 4 4 x= -3 + 7 4 x= -3 + 7 4 x=4 x= -3 - 7 4 x = - 2.5 The Quadratic Formula 2x = x2 - 3 ax2 + bx + c 0 = x2 – 2x - 3 x= 2x = x2 - 3 -2x -2x a = 1 b = -2 ax2 + bx + c -b + √ b2 – 4ac x= 2a x= -(-2) + √ (-2)2 – 4(1)(-3) x= x= 2 2+4 2 x= 2+4 2 x=3 0 = x2 – 2x - 3 c = -3 -(-2) + √ (-2)2 – 4(1)(-3) 2 + √ 4 +12 2 2 + √ 16 2a 2(1) 2(1) x= x= -b + √ b2 – 4ac x= 2-4 2 x = -1 x= 2 + √ 16 2 Solving Quadratic Equations •Graphing •Factoring •Using Square Roots •Completing the Square •Quadratic Formula Solving Quadratic Equations 2 x + 4x - 2 = 0 2 x - 5x + 4 = 0