Hello there! ;) The way you solve this type of problem is you must find common denominators. So simplify 13/3= 4 1/3. So we can do 1/3*5/5=5/15. Next we must do 3/5*3/3= 9/15. So add 5/15+9/15= 14/15. And add the 4+ 14/15= 4 14/15. So then you must subtract 15 - 4 14/15= 10 1/15 I believe. I believe 10 1/15 is the remainder of the rope. You might want to check in with someone else though.
Hope I help! :D
Answer:
FH = 24
Step-by-step explanation:
HG = 13
FH = 37 - 13
= 24
Answer:
![\displaystyle \left(\alpha+1\right)\left(\beta + 1\right) = \frac{a+c-b}{a}\:\: \left(\text{ or } 1+\frac{c-b}{a}\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28%5Calpha%2B1%5Cright%29%5Cleft%28%5Cbeta%20%2B%201%5Cright%29%20%20%3D%20%5Cfrac%7Ba%2Bc-b%7D%7Ba%7D%5C%3A%5C%3A%20%5Cleft%28%5Ctext%7B%20or%20%7D%201%2B%5Cfrac%7Bc-b%7D%7Ba%7D%5Cright%29)
Step-by-step explanation:
We are given the equation:
![ax^2+bx+c=0](https://tex.z-dn.net/?f=ax%5E2%2Bbx%2Bc%3D0)
Which has roots α and β.
And we want to express (α + 1)(β + 1) in terms of <em>a</em>, <em>b</em>, and <em>c</em>.
From the quadratic formula, we know that the two solutions to our equation are:
![\displaystyle x_1 = \frac{-b+\sqrt{b^2-4ac}}{2a}\text{ and } x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x_1%20%3D%20%5Cfrac%7B-b%2B%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%5Ctext%7B%20and%20%7D%20x_2%3D%5Cfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
Let <em>x</em>₁ = α and <em>x₂ </em>= β. Substitute:
![\displaystyle \left(\frac{-b+\sqrt{b^2-4ac}}{2a} + 1\right) \left(\frac{-b-\sqrt{b^2-4ac}}{2a}+1\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28%5Cfrac%7B-b%2B%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%20%2B%201%5Cright%29%20%5Cleft%28%5Cfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%2B1%5Cright%29)
Combine fractions:
![\displaystyle =\left(\frac{-b+2a+\sqrt{b^2-4ac}}{2a} \right) \left(\frac{-b+2a-\sqrt{b^2-4ac}}{2a}\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cleft%28%5Cfrac%7B-b%2B2a%2B%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%20%5Cright%29%20%5Cleft%28%5Cfrac%7B-b%2B2a-%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%5Cright%29)
Rewrite:
![\displaystyle = \frac{\left(-b+2a+\sqrt{b^2-4ac}\right)\left(-b+2a-\sqrt{b^2-4ac}\right)}{(2a)(2a)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%20%5Cfrac%7B%5Cleft%28-b%2B2a%2B%5Csqrt%7Bb%5E2-4ac%7D%5Cright%29%5Cleft%28-b%2B2a-%5Csqrt%7Bb%5E2-4ac%7D%5Cright%29%7D%7B%282a%29%282a%29%7D)
Multiply and group:
![\displaystyle = \frac{((-b+2a)+\sqrt{b^2-4ac})((-b+2a)-\sqrt{b^2-4ac})}{4a^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%20%5Cfrac%7B%28%28-b%2B2a%29%2B%5Csqrt%7Bb%5E2-4ac%7D%29%28%28-b%2B2a%29-%5Csqrt%7Bb%5E2-4ac%7D%29%7D%7B4a%5E2%7D)
Difference of two squares:
![\displaystyle = \frac{\overbrace{(-b+2a)^2 - (\sqrt{b^2-4ac})^2}^{(x+y)(x-y)=x^2-y^2}}{4a^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%20%5Cfrac%7B%5Coverbrace%7B%28-b%2B2a%29%5E2%20-%20%28%5Csqrt%7Bb%5E2-4ac%7D%29%5E2%7D%5E%7B%28x%2By%29%28x-y%29%3Dx%5E2-y%5E2%7D%7D%7B4a%5E2%7D)
Expand and simplify:
![\displaystyle = \frac{(b^2-4ab+4a^2)-(b^2-4ac)}{4a^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%20%5Cfrac%7B%28b%5E2-4ab%2B4a%5E2%29-%28b%5E2-4ac%29%7D%7B4a%5E2%7D)
Distribute:
![\displaystyle = \frac{(b^2-4ab+4a^2)+(-b^2+4ac)}{4a^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%20%5Cfrac%7B%28b%5E2-4ab%2B4a%5E2%29%2B%28-b%5E2%2B4ac%29%7D%7B4a%5E2%7D)
Cancel like terms:
![\displaystyle = \frac{4a^2+4ac-4ab}{4a^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%20%5Cfrac%7B4a%5E2%2B4ac-4ab%7D%7B4a%5E2%7D)
Factor:
![\displaystyle =\frac{4a(a+c-b)}{4a(a)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B4a%28a%2Bc-b%29%7D%7B4a%28a%29%7D)
Cancel. Hence:
![\displaystyle = \frac{a+c-b}{a}\:\: \left(\text{ or } 1+\frac{c-b}{a}\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%20%5Cfrac%7Ba%2Bc-b%7D%7Ba%7D%5C%3A%5C%3A%20%5Cleft%28%5Ctext%7B%20or%20%7D%201%2B%5Cfrac%7Bc-b%7D%7Ba%7D%5Cright%29)
Therefore:
![\displaystyle \left(\alpha+1\right)\left(\beta + 1\right) = \frac{a+c-b}{a}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28%5Calpha%2B1%5Cright%29%5Cleft%28%5Cbeta%20%2B%201%5Cright%29%20%3D%20%5Cfrac%7Ba%2Bc-b%7D%7Ba%7D)
Answer:
y = 4x...
Step-by-step explanation:
Answer
2x
Step-by-step explanation:
-14=20y-7x,10y+4=2x i think