標題:
Quadratic Equations
發問:
1. If the graph of y = 9(x-1)^2 - m - 6 intersects the x-axis, find the smallest value of m. ***2. The expression 2x^2 - mx + 1 has a minimum value -7 for all values of x. Find the value of m.
最佳解答:
1. If the graph of y = 9(x-1)^2 - m - 6 intersects the x-axis, find the smallest value of m. y = 9(x - 1)^2 - m - 6 y = 9(x^2 - 2x + 1) - m - 6 y = 9x^2 - 18x + 9 - m - 6 y = 9x^2 - 18x + 3 - m Intersecting the x-axis there is real roots for y = 9x^2 - 18x + 3 - m = 0 Discriminant >= 0 18^2 - (4)(9)(3 - m) >= 0 324 - 36(3 - m) >= 0 9 >= 3 - m m >= -6 The smallest value of m is -6 2. The expression 2x^2 - mx + 1 has a minimum value -7 for all values of x. Find the value of m. 2x^2 - mx + 1 = 2(x^2 - mx/2 + 1/2) = 2[x^2 - mx/2 + (m/4)^2 +1/2 - (m/4)^2] = 2(x - m/4)^2 + 1 - m^2/8 The minimum value happens when the braketed terms is zero, and the minimum value is 1 - m^2/8 = -7 8 = m^2/8 m^2 = 64 m = +/-8
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