9-6 The Quadratic Formula and Discriminant

Writing Quadratic Equations in Standard Form ax² + bx + c = 0 a, b, and c should be on the same side of the equal sign a should always be positive Ex. 3x² + 2x = -5 3x² + 2x + 5 = 0

If ax² + bx + c = 0 and a ≠ 0, then 𝒙= −𝒃± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂 Quadratic Formula If ax² + bx + c = 0 and a ≠ 0, then 𝒙= −𝒃± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂

Solving using the Quadratic Formula Use ax² +bx + c = 0 to substitute a, b, and c into 𝒙= −𝒃± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂 **caution** if ‘b’ is negative (ax² - bx + c + 0), you will still need to use the negative sign from the equation. You will end up with a double negative (which will make ‘b’ positive) **if ‘b’ is negative then: 𝑥= −−𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 = 𝑥= 𝑏± 𝑏 2 −4𝑎𝑐 2𝑎

Examples a) x² - 8 = 2x b) 2x² + 5x + 3 = 0

Discriminant The expression under the radical of the quadratic formula b² - 4ac

Real Solutions of the Discriminant # of Real Solutions Graph > 0   = 0 < 0

Examples a) x² + 4x – 15 = 0 b) 3x² - 7x = -5

Homework Page 279 36-41