E Quadratic Formula Discriminant Objective To solve quadratic
E) Quadratic Formula & Discriminant Objective: To solve quadratic equations using the quadratic formula and calculate the number of solutions using the discriminant.
Quadratic Formula and Discriminant
Discriminant: Used to determine the # of x-intercepts real b 2 − 4 ac = + Two roots real b 2 − 4 ac = 0 One root b 2 No real − 4 ac = − roots No Solution
Use the Discriminant to find the # of roots then graph the equation to find the x-intercepts 1) y = x 2 + − 0 x 4 Discriminant b 2 – 4 ac = (0)2 – 4(1)(-4) = + Axis of Symmetry x = -b = −( 0 ) =0 2 a 2( 1 ) Vertex y = ( 0 )2 − 4 = -4 2 nd Point Choose x = 1 y = ( 1 )2 − 4 = -3 x = {-2, 2}
Use the Discriminant to find the # of roots then graph the equation to find the x-intercepts 2) y = x 2 + 0 x + 0 Discriminant b 2 – 4 ac = (0)2 – 4(1)(0 ) = 0 Axis of Symmetry x = -b = −( 0 ) =0 2 a 2( 1 ) Vertex y = ( 0 )2 = 0 2 nd Point Choose x = 1 y = ( 0 )2 = 0 x=0
Use the Discriminant to find the # of roots then graph the equation to find the x-intercepts 3) y = x 2 + 0 x 4 Discriminant b 2 – 4 ac = (0)2 – 4(1)(4 ) = − Axis of Symmetry x = -b = −( 0 ) =0 2 a 2( 1 ) Vertex y = ( 0 )2 + 4 = 4 2 nd Point Choose x = 1 y = ( 1 )2 + 4 = 3 No x-intercepts
Quadratic Formula Used to find the x-intercepts Axis of Symmetry Discriminant Play Video x = -b ± 2 a Rules 1) Write problem as a quadratic equation (ax 2 +bx + c = 0) 2) Determine values for a, b, and c 3) Plug values into the Quadratic Formula 4) Simplify the Discriminant (factor out “pairs” if possible) 5) Separate into 2 problems (+) and (−) and solve
Example 1 y = -x 2 + 5 x + 2 a b c - (5 ) ± √( 5 )2 − 4(-1)( 2 ) = 2(-1) =
Example 2 y = 2 x 2 + 5 x − 3 a b c - (5 ) ± √( 5 )2 − 4( 2 )(-3) 2( 2 )
Example 4 x 2 + 3 x = 7 1 a b c - ( 3 ) ± √( 3 )2 − 4( 1 )(-7) = 2( 1 ) =
Example 5 y = -x 2 + 4 x − 7 a b c - (4 ) ± √( 4 )2 − 4(-1)(-7) = 2(-1) = No Solution
Example 6 Solve a b a= 1 b = -6 c= 9 c = = = 2
Classwork 1) 2) y = 2 x 2 − x + 1 x = 3, 1 2 No Solution 3) y = x 2 − x − 2 4) y = x 2 − 2 x + 1 x = {-1, 2} 5) y = 3 x 2 − 2 x + 4 No Solution x=1 6) y = -x 2 + 5 x + 1
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