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Transcript
Solutions to quadratic equations 2x2+7x+6 (2A.8) Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: (B) analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations using the quadratic formula; The Babylonians Stopping a car Every polynomial has as many roots as its degree A quadratic equation is a second degree polynomial, it has two roots x2 − 4 20 15 10 5 -4 -2 2 4 x2 − 4 20 15 10 5 -4 -2 2 (x+2)(x−2) 4 x2 +2x +1 25 20 15 10 5 -4 -2 2 4 x2 +2x +1 25 20 15 10 5 -4 -2 2 (x+1)2 4 x2 +2x +5 25 20 15 10 5 -4 -2 2 4 x2 +2x +5 25 20 15 10 5 -4 -2 2 4 (x+1+2i)(x+1−2i) ax2+bx+c=0 The quadratic formula works for any quadratic equation because it has already done the steps of completing the square for you. All you have to do to use it is substitute the coefficients into the formula and simplify. x = −b±√(b2 − 4ac) 2a 1. Get the equation in ax2+bx+c=0 form 2. Label the coefficients a,b and c then substitute into the formula and simplifiy. x = −b±√(b2 − 4ac) 2a 4x2+x=3 100 80 60 40 20 -4 -2 2 4 4x2+x−3=0 a=4, b=1, c=−3 4x2+x−3=0 a=4, b=1, c=−3 x=−1±√12 − 4(4)(−3) 2(4) 4x2+x−3=0 a=4, b=1, c=−3 x=−1±√1 + 48 8 4x2+x−3=0 a=4, b=1, c=−3 x=−1±√49 8 4x2+x−3=0 a=4, b=1, c=−3 x=−1±7 8 4x2+x−3=0 a=4, b=1, c=−3 x=−1+7 x=−1−7 8 8 4x2+x−3=0 a=4, b=1, c=−3 x=6 x=−8 8 8 4x2+x−3=0 a=4, b=1, c=−3 x=3 x=−1 4 9x2 −30x+25=0 250 200 150 100 50 -4 -2 2 4 9x2 −30x+25=0 a=9, b=−30, c=25 9x2 −30x+25=0 a=9, b=−30, c=25 x=30±√302 − 4(9)(25) 2(9) 9x2 −30x+25=0 a=9, b=−30, c=25 x=30±√900 − 900 18 9x2 −30x+25=0 a=9, b=−30, c=25 x=30±0 18 9x2 −30x+25=0 a=9, b=−30, c=25 x=30 18 9x2 −30x+25=0 a=9, b=−30, c=25 x=5 3 Exercises – solve using the quadratic formula 1. x2+5x+6=0 2. x2−6x=−8 3. 2x2+3x=0 4. 2x2−7x−4=0 5. x2+x=12 6. 3x2+x−2=0 7. x2−9=0 8. x2+2x−8=0 To memorize the quadratic formula for musical intelligences put these words to Pop Goes the Weasel Minus bee plu-us or minus the square root of bee squared minus four a-a-ay cee all over two ay. For bodily-kinesthetic learners write the quadratic formula before each homework problem Before a test read the quadratic formula before going into class write it down when you first get the test A Visual Quadratic Equation x2 + − x = ± x+ =0 √ − The discriminant tells you how many and what type of solutions a quadratic equation has. A quadratic equation is a second degree equation so it can have at most two solutions The discriminant is b2 − 4ac How many solutions did you get, what type of solutions were they and what was the sign of the discriminant in the equation 4x2+x=3 How many solutions did you get, what type of solutions were they and what was the sign of the discriminant in the equation 9x2 −30x+25=0 How many solutions did you get, what type of solutions were they and what was the sign of the discriminant in the equation x2 −3x=−10 Equation Type 4x2+x=3 real Discriminant #Solutions + 2 9x2 −30x+25=0 real 0 1 x2 −3x=−10 complex − 2 If the discriminant is positive you get two real solutions If the discriminant is 0 you get one real solution If the discriminant is negative you get two complex solutions Exercises – describe the solutions using the discriminant 1. 4x2−20x+25=0 2. 5x2−6x=-2 3. 10x2=x+2 4. x2+x−1=0