9.6.2 How many solutions? How many times does the parabola cross the x-­‐axis? Two, one, or none? How can you tell without graphing? ex1) x2 = 16, 2 ex2) x2 = -­‐81, 0 ex3) x2 = 0, 1 Part of the quadratic formula can tell you the same thing. That part is called the discriminant. −𝑏 ± 𝑏 ! − 4𝑎𝑐
𝑥 = 2𝑎
If 𝑏 ! − 4𝑎𝑐 > 0, then there are two real solutions If 𝑏 ! − 4𝑎𝑐 < 0, then there are no real solutions If 𝑏 ! − 4𝑎𝑐 = 0, then there is one real solution ex 4) How may real-­‐number solutions does x2 + 3x + 11 = 0? a = 1, b = 3, c = 11 𝑏 ! − 4𝑎𝑐 (3)2 -­‐ 4(1)(11) 9 -­‐ 44 -­‐35, so no real solutions What method (quadratic formula, factoring, graphing, square roots) would you choose to solve? ex 5) x2 -­‐ 81 = 0 +81 +81 x2 = 81 x = -­‐9 or x = 9 ex 6) -­‐3x2 -­‐ 11x + 4 = 0 a=-­‐3, b=-­‐11, c=4 −(−11) ± (−11)! − 4(−3)(4)
𝑥 = 2(−3)
11 ± 121 + 48
𝑥 = −6
11 ± 169
𝑥 = −6
11 ± 13
𝑥 = −6
24
−2
𝑜𝑟 𝑥 = −6
−6
1
𝑥 = −4 𝑜𝑟 𝑥 = 3
𝑥 = 607/19-­‐27, 29, 31, 41