Complete the remaining problems showing all your work Given

Complete the remaining problems, showing all your work Given that f(x) = 5x^2 + 50x + 123 1/5 has a squareroot at x -4 2/5 use the axis of symmetry to find the other squareroot of the function. Solve 5w^2 - 4w = -2 3/5 by completing the square. Simply each expression (write any complex answer in standard form) (7 = i) + (4 0- 3i) (-3-3i) - (-4 - (-3)i)

Solution

11. y = f (x) = 5x² + 50x + 616/5. We know that for a quadratic function in standard form, y=ax2+bx+c, the axis of symmetry is a vertical line x=b/2a. Here, a = 5, b = 50 so that the axis of symmetry is the vertical line x = - 50/10 i.e. x = -5 Since the given equation has a root at x = - 22/5, and since the given parabola is symmetric about its line of symmetry, the other root of the quadratic function will have the same distance from the line of symmetry as the known root which is at a distance of   5 – 22/5= 3/5 units from the line of symmetry. Thus, the other root of the given function is x = -5 -3/5 = -28/5.

12. 5w² -4w= - 13/5 or, 25w² – 20w + 13 = 0 or, (5w)² -2(5w)(2)+ 4 + 9 = 0 or, (5w – 2)² = - 9 so that    5w – 2 = ± -9 or, 5w – 2 = ± 3i. Then 5w = 2 ± 3i so that w = 2/5 ± 3i/5.

13. ( 7 + i) + ( 4 – 3i) = 7 + 4 + i( 1- 3) = 11 -2i + 11 + i( -2).

14. (-3 – 4i) – ( -4 + 3i) = -3 + 4 – 4i -3i = 1 – 7i = 1 + i(-7)

15. ( 6-2i)(-4 + 5i) = -24 + 8i + 30i -10i² = -24 + 8i + 30i + 10 = -14 +38i ( as i² = -1)

16. (7 -3i)/( -2 -5i)=(7 -3i)(-2+5i)/ (-2 -5i)(-2 +5i) = ( -14 + 6i + 35i -15i²)/ [ (-2)²– (-5i)²] = ( -14+15 + 41i)/[ 4 – 25i²] = (1 + 41i)/ ( 4 + 25) = (1 + 41i)/29 = 1/29 + (41/29)i