Solve the quadratic equation; 12y^2 + 29y + 15 = 0

Answer 1

Can use the "ac" method; ay² + by + c = 0

Here a = 12; b = 29; c = 15, so ac = 180. There are no common factors with which to reduce the coefficients (29 is a prime no.)

We need factors of 180 that combine via addition to make 29. It's addition because all of the coefficients are positive.

Factors of 180 are: 180,1; 90,2; 60,3; 45,4; 5,36; 6,30; 9,20) Aha! 9 + 20 = 29!

So let's expand the original eqn: 12y² + 9y + 20y + 15 = 0 (It doesn't matter if you reverse the order of the new terms - it'll still work). Now factor: 3y(4y + 3) + 5(4y + 3) = 0. Notice that 2 of the factors are identical, so they can be factored out. (4y + 3)(3y + 5) = 0. Now can set each = 0 to solve for y.

4y + 3 = 0; y = -3/4

3y + 5 = 0; y = -5/3

Answer 2

12y² +29y +15 = 0

(3y + 5)(4y + 3) = 0

3y+5 = 0 or 4y+3 = 0

y = -5/3 or y = -3/4

Answer 3

 Here the most straightforward way is to use the quadratic formula

y_1,2 = (-b ± sqrt(b²-4ac)) / 2a,

where

a = 12

b = 29

c = 15

plugging in the numbers gives

y_1,2 = (-29 ± sqrt(29²-4*12*15)) / (2*12) = (-29 ± 11) / 24

y₁ = (- 29 - 11) / 24 = - 40 / 24 = - 5/3

y₂ = (- 29 + 11) / 24 = - 18 / 24 = - 3/4

We can then check our work by either substituting the values into the original equation or by using the Vieta's formulas (en.wikipedia.org/wiki/Vieta%27s_formulas#Example) as follows

y₁ + y₂ = - 5/3 - 3/4 = - 20 / 12 - 9 / 12 = - 29 / 12 = - b / a

y₁ × y₂ = (- 5/3) × (- 3/4) = 5 / 4 = 15 / 12 = c / a