Answer: 1.3125
Step-by-step explanation: Rise over run so 21/16, which if you open up your calculator you can divide 21 by 16 and your answer is 1.3125. Have a great day and message me with any more questions!
Standard Form: 12x+5y=-10
Answer:
$7,500
Step-by-step explanation:
2,500 - 1000 = 1500
1500 * 5 = 7,500
Answer:
26. 525198938992
Step-by-step explanation:
Uhmmmmmmmmmm
Answer:
<em>a) </em>
<em />
<em />
<em>b) p is in the interval (-4,4)</em>
Step-by-step explanation:
<u>Quadratic Equation</u>
It's given the following quadratic equation:
![x^2+px+4=0](https://tex.z-dn.net/?f=x%5E2%2Bpx%2B4%3D0)
a)
It's required to complete squares and find the roots. This can be done by recalling the polynomial identity:
![a^2+2ab+b^2=(a+b)^2](https://tex.z-dn.net/?f=a%5E2%2B2ab%2Bb%5E2%3D%28a%2Bb%29%5E2)
We already have the first term squared, and we need to find the second term. Rewriting the equation:
![\displaystyle x^2+2(\frac{1}{2}px)+4=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%5E2%2B2%28%5Cfrac%7B1%7D%7B2%7Dpx%29%2B4%3D0)
The second term of the binomial is 1/2p, thus completing the squares with
:
![\displaystyle x^2+2(\frac{1}{2}px)+\left(\frac{1}{2}p\right)^2+4-\left(\frac{1}{2}p\right)^2=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%5E2%2B2%28%5Cfrac%7B1%7D%7B2%7Dpx%29%2B%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dp%5Cright%29%5E2%2B4-%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dp%5Cright%29%5E2%3D0)
Factoring:
![\displaystyle \left(x+\frac{1}{2}p\right)^2+4-\frac{1}{4}p^2=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28x%2B%5Cfrac%7B1%7D%7B2%7Dp%5Cright%29%5E2%2B4-%5Cfrac%7B1%7D%7B4%7Dp%5E2%3D0)
Moving the independent term to the right side:
![\displaystyle \left(x+\frac{1}{2}p\right)^2=\frac{1}{4}p^2-4](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28x%2B%5Cfrac%7B1%7D%7B2%7Dp%5Cright%29%5E2%3D%5Cfrac%7B1%7D%7B4%7Dp%5E2-4)
Taking the square root:
![\displaystyle x+\frac{1}{2}p=\pm\sqrt{\frac{1}{4}p^2-4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%2B%5Cfrac%7B1%7D%7B2%7Dp%3D%5Cpm%5Csqrt%7B%5Cfrac%7B1%7D%7B4%7Dp%5E2-4%7D)
Solving for x:
![\displaystyle x=-\frac{1}{2}p\pm\sqrt{\frac{1}{4}p^2-4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%3D-%5Cfrac%7B1%7D%7B2%7Dp%5Cpm%5Csqrt%7B%5Cfrac%7B1%7D%7B4%7Dp%5E2-4%7D)
b) If the equation won't have real roots, then the radicand should be negative:
![\displaystyle \frac{1}{4}p^2-4](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B4%7Dp%5E2-4%3C0)
Factoring:
![\displaystyle \left(\frac{1}{2}p-2\right)\left(\frac{1}{2}p+2\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dp-2%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dp%2B2%5Cright%29%3C0)
The zeros of the left-side polynomial are:
![\displaystyle \frac{1}{2}p-2=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2%7Dp-2%3D0)
![\displaystyle \frac{1}{2}p=2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2%7Dp%3D2)
p = 4
![\displaystyle \frac{1}{2}p+2=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2%7Dp%2B2%3D0)
![\displaystyle \frac{1}{2}p=-2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2%7Dp%3D-2)
p = -4
The inequality:
![\displaystyle \left(\frac{1}{2}p-2\right)\left(\frac{1}{2}p+2\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dp-2%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dp%2B2%5Cright%29%3C0)
Is satisfied for values of p in the interval (-4,4)