Question

Determine the zeros) of the quadratic function y=6x^2 + 13x + 5. Show all steps. (

Answer 1

We solve 6x^2 + 13x + 5 = 0.

1) Try factoring. Multiply a·c = 6·5 = 30. Find two integers that multiply to 30 and add to 13: 10 and 3.

2) Split the middle term using 10x and 3x:
6x^2 + 10x + 3x + 5

3) Group and factor:
(6x^2 + 10x) + (3x + 5)
= 2x(3x + 5) + 1(3x + 5)
= (3x + 5)(2x + 1)

4) Set each factor to zero:
3x + 5 = 0 => x = -5/3
2x + 1 = 0 => x = -1/2

(You can check with the quadratic formula: x = [-13 ± sqrt(13^2 - 4·6·5)]/(2·6) = [-13 ± 7]/12 gives x = -1/2 and x = -5/3.)

Zeros: x = -5/3 and x = -1/2 (points on the graph: (-5/3, 0) and (-1/2, 0)).