Respuesta :
Answer:
Jennifer would complete the job on her own in 5 hours.
Step-by-step explanation:
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
[tex]ax^{2} + bx + c, a\neq0[/tex].
This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})[/tex], given by the following formulas:
[tex]x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}[/tex]
[tex]\Delta = b^{2} - 4ac[/tex]
Rates
The together rate is the sum of their separates rate.
We have that:
Jennifer takes x hours to complete the job on her own, so her rate is 1/x.
Carl needs 15 hours longer than Jennifer, that is, 15 + x hours, so his rate is 1/(15+x).
The together rate is 1/4. So
[tex]\frac{1}{4} = \frac{1}{x} + \frac{1}{15+x}[/tex]
[tex]\frac{1}{4} = \frac{15 + x + x}{x(15+x)}[/tex]
[tex]\frac{1}{4} = \frac{15 + 2x}{x(15+x)}[/tex]
Applying cross multiplication.
[tex]x^2 + 15x = 4(15 + 2x)[/tex]
[tex]x^2 + 15x = 60 + 8x[/tex]
[tex]x^2 + 7x - 60 = 0[/tex]
Quadratic equation with [tex]a = 1, b = 7, c = -60[/tex]. So
[tex]\Delta = 7^{2} - 4*1*60 = 289[/tex]
[tex]x_{1} = \frac{-7 + \sqrt{289}}{2} = 5[/tex]
[tex]x_{2} = \frac{-7 - \sqrt{289}}{2} = -12[/tex]
Since the time to complete the job has to be a positive value.
Jennifer would complete the job on her own in 5 hours.
