Solve for roots with quadratic equation and the vertex for

Question

y= -5x² + 9x - 11? 

Answer 1

y = - 5x² + 9x - 11

To find roots substitute y = 0 in above equation.

0 = - 5x² + 9x - 11

 5x² - 9x + 11 = 0

Compare it to quadratic equation ax² + bx + c = 0

a = 5, b = - 9 and c = 11

Discriminant = b² - 4ac = (- 9)² - 4(5)(11)

= 81 - 220 = - 139

Discriminant is nagative, then the roots are imaginary.

Roots are x = [ - b ± √(b² - 4ac)]/2a

x = [9 ± √(- 139)]/10

x = [9 ± 11.78i]/10

x = [9 + 11.78i]/10 and x = [9 - 11.78i]/10

Roots are x = 0.9 + 1.178i and x = 0.9 - 1.178i

 

Now find the vertex of y = - 5x² + 9x - 11.

Compare it to quadratic function y = ax² + bx + c.

a = - 5, b =  9 and c = - 11

 x coordinate of vertex x = - b/2a

x = - (9)/[2(- 5)]

x = 9/10

x =  0.9

To find y coordinate of vertex substitute x =  0.9 in y = - 5x² + 9x - 11.

y = - 5(0.9)² + 9( 0.9) - 11

y = - 4.05 + 8.1 - 11

y = - 6.95

Vertex is (x, y) = ( 0.9, - 6.95).