Answer:
12
Step-by-step explanation:
A midpoint of a line means that each segment connecting from the midpoint to an end is equal. For this problem, this means that AB = BC, as B is the midpoint, and A and B are the ends. Therefore, we can say that:
AB = BC
2x + 6 = 5x - 3
add 3 to both sides
2x + 9 = 5x
subtract 2x from both sides
9 = 3x
divide both sides by 3
3 = x
Plugging 3=x into AB, this means that 2(3) + 6 = AB = 12
Answer:
![l \approx 15.455\,ft](https://tex.z-dn.net/?f=l%20%5Capprox%2015.455%5C%2Cft)
Step-by-step explanation:
The length of the ramp is computed with the help of the following trigonometric function:
![\sin \theta = \frac{h}{l}](https://tex.z-dn.net/?f=%5Csin%20%5Ctheta%20%3D%20%5Cfrac%7Bh%7D%7Bl%7D)
![l = \frac{h}{\sin \theta}](https://tex.z-dn.net/?f=l%20%3D%20%5Cfrac%7Bh%7D%7B%5Csin%20%5Ctheta%7D)
![l = \frac{4\,ft}{\sin 15^{\circ}}](https://tex.z-dn.net/?f=l%20%3D%20%5Cfrac%7B4%5C%2Cft%7D%7B%5Csin%2015%5E%7B%5Ccirc%7D%7D)
![l \approx 15.455\,ft](https://tex.z-dn.net/?f=l%20%5Capprox%2015.455%5C%2Cft)
![\huge \boxed{\mathfrak{Question} \downarrow}](https://tex.z-dn.net/?f=%20%5Chuge%20%5Cboxed%7B%5Cmathfrak%7BQuestion%7D%20%5Cdownarrow%7D)
Solve the equation using the quadratic formula ⇨ x² + 11x + 9 = 0
![\large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}](https://tex.z-dn.net/?f=%20%5Clarge%20%5Cboxed%7B%5Cmathfrak%7BAnswer%20%5C%3A%20with%20%5C%3A%20Explanation%7D%20%5Cdownarrow%7D)
![\sf \: x ^ { 2 } + 11 x + 9 = 0](https://tex.z-dn.net/?f=%20%5Csf%20%5C%3A%20x%20%5E%20%7B%202%20%7D%20%2B%2011%20x%20%2B%209%20%3D%200)
All equations of the form
can be solved using the quadratic formula:
. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
![\sf \: x^{2}+11x+9=0](https://tex.z-dn.net/?f=%20%5Csf%20%5C%3A%20x%5E%7B2%7D%2B11x%2B9%3D0%20)
This equation is in standard form: ax² + bx + c = 0. Substitute 1 for a, 11 for b and 9 for c in the quadratic formula
.
![\sf \: x=\frac{-11±\sqrt{11^{2}-4\times 9}}{2} \\](https://tex.z-dn.net/?f=%20%5Csf%20%5C%3A%20x%3D%5Cfrac%7B-11%C2%B1%5Csqrt%7B11%5E%7B2%7D-4%5Ctimes%209%7D%7D%7B2%7D%20%20%5C%5C%20)
Square 11.
![\sf \: x=\frac{-11±\sqrt{121-4\times 9}}{2} \\](https://tex.z-dn.net/?f=%20%5Csf%20%5C%3A%20x%3D%5Cfrac%7B-11%C2%B1%5Csqrt%7B121-4%5Ctimes%209%7D%7D%7B2%7D%20%20%5C%5C%20)
Multiply -4 times 9.
![\sf \: x=\frac{-11±\sqrt{121-36}}{2} \\](https://tex.z-dn.net/?f=%20%5Csf%20%5C%3A%20x%3D%5Cfrac%7B-11%C2%B1%5Csqrt%7B121-36%7D%7D%7B2%7D%20%20%5C%5C%20)
Add 121 to -36.
![\sf \: x=\frac{-11±\sqrt{85}}{2} \\](https://tex.z-dn.net/?f=%20%5Csf%20%5C%3A%20x%3D%5Cfrac%7B-11%C2%B1%5Csqrt%7B85%7D%7D%7B2%7D%20%20%5C%5C%20)
Now solve the equation
when ± is plus. Add -11 to √85.
![\boxed{ \boxed{\bf \: x=\frac{\sqrt{85}-11}{2} }}](https://tex.z-dn.net/?f=%20%5Cboxed%7B%20%20%5Cboxed%7B%5Cbf%20%5C%3A%20x%3D%5Cfrac%7B%5Csqrt%7B85%7D-11%7D%7B2%7D%20%7D%7D)
Now solve the equation
when ± is minus. Subtract √85 from -11.
![\boxed{ \boxed{\bf \: x=\frac{-\sqrt{85}-11}{2}} } \\](https://tex.z-dn.net/?f=%20%20%5Cboxed%7B%20%5Cboxed%7B%5Cbf%20%5C%3A%20x%3D%5Cfrac%7B-%5Csqrt%7B85%7D-11%7D%7B2%7D%7D%20%7D%20%5C%5C%20)
The equation is now solved. The solution set is :-
![\bf \: x=\frac{\sqrt{85}-11}{2} \\ \\ \sf \: and \\ \\ \bf \: x=\frac{-\sqrt{85}-11}{2}](https://tex.z-dn.net/?f=%5Cbf%20%5C%3A%20x%3D%5Cfrac%7B%5Csqrt%7B85%7D-11%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20%20%5Csf%20%5C%3A%20and%20%5C%5C%20%20%5C%5C%20%20%20%5Cbf%20%5C%3A%20x%3D%5Cfrac%7B-%5Csqrt%7B85%7D-11%7D%7B2%7D%20)
Answer: -5, -2.5, 0, 2.5, 5, 7.5
The answer fits in with all the requirements and is the only one that starts with -5.
Hope this helped
I’m just doing this to ask more questions