To find the value of ‘a’ in the equation a² + 3a – 8 = 0, we can use the quadratic formula:
a = rac{{-b
ightarrow ext{±} ext{sqrt}{{b^2 – 4ac}}}}{2a}
In our equation, we identify the coefficients:
- a = 1 (the coefficient of a²)
- b = 3 (the coefficient of a)
- c = -8 (the constant term)
Plugging these values into the quadratic formula gives us:
a = rac{{-3 ext{±} ext{sqrt}{{3^2 – 4 imes 1 imes (-8)}}}}{2 imes 1}
Now, we calculate the discriminant:
b^2 – 4ac = 3^2 – 4(1)(-8) = 9 + 32 = 41
This means the equation simplifies to:
a = rac{{-3 ext{±} ext{sqrt}(41)}}{2}
Now we can calculate the two possible values of ‘a’:
a = rac{{-3 + ext{sqrt}(41)}}{2} ext{ or } a = rac{{-3 – ext{sqrt}(41)}}{2}
Approximating the square root, we find:
sqrt(41) ext{ is approximately } 6.4
Thus our solutions become:
a = rac{{-3 + 6.4}}{2} = rac{3.4}{2} = 1.7
And:
a = rac{{-3 – 6.4}}{2} = rac{-9.4}{2} = -4.7
So, the two possible values for ‘a’ are:
a ≈ 1.7 ext{ or } a ≈ -4.7
In conclusion, the values of ‘a’ that satisfy the equation a² + 3a – 8 = 0 are approximately 1.7 and -4.7.