6y^2 +13y - 8
19y^2-8
6y^2+13y-8 = (2y-1)(3y+8)
19y^2-8 = (√(19)y-√8)(√(19)y+8)
To solve the expression 6y^2 + 13y - 8, you can use the quadratic formula or factorization. Let's start with factorization.
First, we need to find two numbers that multiply to give -48 (the product of the coefficient of y^2 and the constant term) and add to give the coefficient of y (in this case, 13).
Let's list the factors of -48:
-1, -2, -3, -4, -6, -8, -12, -16, -24, -48
and try adding pairs of them to find the combination that adds up to 13:
-1 + (-48) = -49
-2 + (-24) = -26
-3 + (-16) = -19
-4 + (-12) = -16
-6 + (-8) = -14
None of these combinations give us the desired sum of 13. So factorization may not be possible in this case.
Next, let's solve the expression using the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, where a, b, and c are coefficients, the solutions can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For the expression 6y^2 + 13y - 8, we have:
a = 6
b = 13
c = -8
Plugging these values into the quadratic formula, we get:
y = (-13 ± √(13^2 - 4(6)(-8))) / 2(6)
Simplifying further:
y = (-13 ± √(169 + 192)) / 12
y = (-13 ± √(361)) / 12
y = (-13 ± 19) / 12
Breaking it down into two cases:
Case 1: y = (-13 + 19) / 12 = 6 / 12 = 1/2
Case 2: y = (-13 - 19) / 12 = -32/12 = -8/3
Therefore, the solutions to the expression 6y^2 + 13y - 8 are y = 1/2 and y = -8/3.