A quadratic equation that has 2 roots that are equal means that the discriminant of the equation is 0. The value of k that makes the equation have 2 real and equal roots is ![k = \frac{5}{6}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B5%7D%7B6%7D)
Given that:
![(2k + 1)x^2 + 2x = 10x - 6](https://tex.z-dn.net/?f=%282k%20%2B%201%29x%5E2%20%2B%202x%20%3D%2010x%20-%206)
Rewrite as:
![(2k + 1)x^2 + 2x - 10x + 6 = 0](https://tex.z-dn.net/?f=%282k%20%2B%201%29x%5E2%20%2B%202x%20-%2010x%20%2B%206%20%3D%200)
![(2k + 1)x^2 - 8x + 6 = 0](https://tex.z-dn.net/?f=%282k%20%2B%201%29x%5E2%20-%208x%20%2B%206%20%3D%200)
A quadratic equation is represented as:
![ax^2 + bx + c = 0](https://tex.z-dn.net/?f=ax%5E2%20%2B%20bx%20%2B%20c%20%3D%200)
By comparison:
![a=2k+1\\ b =-8\\\ c =6](https://tex.z-dn.net/?f=a%3D2k%2B1%5C%5C%20b%20%3D-8%5C%5C%5C%20c%20%3D6)
If an equation has 2 real and equal roots, then:
![b^2 = 4ac](https://tex.z-dn.net/?f=b%5E2%20%3D%204ac)
So, we have:
![(-8)^2 = 4 \times (2k + 1) \times 6](https://tex.z-dn.net/?f=%28-8%29%5E2%20%3D%204%20%5Ctimes%20%282k%20%2B%201%29%20%5Ctimes%206)
![64 = 4 \times (2k + 1) \times 6](https://tex.z-dn.net/?f=64%20%3D%204%20%5Ctimes%20%282k%20%2B%201%29%20%5Ctimes%206)
Divide by 4
![16 = (2k + 1) \times 6](https://tex.z-dn.net/?f=16%20%3D%20%282k%20%2B%201%29%20%5Ctimes%206)
Divide by 6
![\frac{16}{6} = (2k + 1)](https://tex.z-dn.net/?f=%5Cfrac%7B16%7D%7B6%7D%20%3D%20%282k%20%2B%201%29)
![\frac{8}{3} = 2k + 1](https://tex.z-dn.net/?f=%5Cfrac%7B8%7D%7B3%7D%20%3D%202k%20%2B%201)
Subtract 2 from both sides
![2k = \frac{8}{3} -1](https://tex.z-dn.net/?f=2k%20%3D%20%5Cfrac%7B8%7D%7B3%7D%20-1)
Take LCM
![2k = \frac{8-3}{3}](https://tex.z-dn.net/?f=2k%20%3D%20%5Cfrac%7B8-3%7D%7B3%7D)
![2k = \frac{5}{3}](https://tex.z-dn.net/?f=2k%20%3D%20%5Cfrac%7B5%7D%7B3%7D)
Divide by 2
![k = \frac{5}{6}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B5%7D%7B6%7D)
Hence, the value of k that makes the equation have 2 real and equal roots is ![k = \frac{5}{6}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B5%7D%7B6%7D)
Read more about real and equal roots at:
brainly.com/question/2535040
The slope of that line is “undefined”. The slope of horizontal lines are 0 and the slope of vertical lines are undefined.
Answer:
A
Step-by-step explanation:
A line segment is represented by end points on each end of the line segment. A line in geometry is represented by a line with arrows at each end.
Power of a power as applied to exponents is the following:
The power of a power is the product of the exponents, or
(x^a)^b = x^(ab)
For example
(5^2)^3=5^(2*3)=5^6=15625