5.6 Quadratic Formula & Discriminant

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Presentation transcript:

5.6 Quadratic Formula & Discriminant

Quadratic Formula (Yes, it’s the one with the song!) If you take a quadratic equation in standard form (ax2+bx+c=0), and you complete the square, you will get the quadratic formula!

When to use the Quadratic Formula Use the quadratic formula when you can’t factor to solve a quadratic equation (or when you’re stuck on how to factor the equation).

Examples 1. 3x2 + 8x = 35 3x2 + 8x – 35 = 0 a = 3, b = 8, c = -35 OR

2. -2x2 = -2x + 3 -2x2 + 2x – 3 =0 a = -2, b = 2, c = -3

3. 4x2 = 20x - 25 4x2 - 20x + 25 = 0 a = 4, b = -20, c = 25

Discriminant: b2-4ac The discriminant tells you how many solutions and what type you will have. If the discriminant: Is positive – 2 real solutions Is negative – 2 imaginary solutions Is zero – 1 real solution

Examples 9x2+6x-4=0 a=9, b=6, c=-4 b2-4ac=(6)2-4(9)(-4) =36+144=180 2 real solutions c. 9x2+6x+5=0 a=9, b=6, c=5 b2-4ac=(6)2-4(9)(5) =36-180=-144 2 imaginary solutions Find the discriminant and give the number and type of solutions. 9x2+6x+1=0 a=9, b=6, c=1 b2-4ac=(6)2-4(9)(1) =36-36=0 1 real solution

Deriving the Quadratic Formula by Completing the Square ax2 + bx + c = 0 Standard Form ax2 + bx = -c Isolate x’s on one side Divide each term by a Complete the square Write as a binomial square Find common denominator

The Quadratic Formula!! Combine fractions on the right Take the square root of both sides Isolate the x Separate the radical Simplify the denominator Combine the fractions