Answer:
2 solutions
Step-by-step explanation:
Given
2x² + 3x - 5 ( to solve the quadratic equate it to zero )
2x² + 3x - 5 = 0 ← in standard form
(2x + 5)(x - 1) = 0 ← in factored form
Equate each factor to zero and solve for x
2x + 5 = 0 ⇒ 2x = - 5 ⇒ x = - ![\frac{5}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B2%7D)
x - 1 = 0 ⇒ x = 1
The 2 solutions are x = -
, x = 1
I believe the correct answer is 2,097,152 because:
2x4= 8 (2)
8x4= 32 (3)
32x4= 128 (4)
128x4= 512 (5)
512x4= 2,048 (6)
2,048x4= 8,192 (7)
8,192x4= 32,768 (8)
131,072x4= 524,288 (9)
524,288x4= 2,097,152 (10)
Answer:
Center = (7, - 5)
Radius = 3
Step-by-step explanation:
![{x}^{2} + {y}^{2} + 14x - 10y + 65 = 0 \\ {x}^{2} + 14x + {y}^{2} - 10 + 65 = 0 \\ ( {x}^{2} + 14x + {7}^{2}) - {7}^{2} + ({y}^{2} - 10y + {5}^{2}) \\- {5}^{2} + 65 = 0 \\ (x + 7)^{2} + (y - 5)^{2} - 49 - 25 + 65 = 0 \\ (x + 7)^{2} + (y - 5)^{2} - 49 + 40= 0 \\ (x + 7)^{2} + (y - 5)^{2} - 9= 0 \\ (x + 7)^{2} + (y - 5)^{2} = 9 \\ (x + 7)^{2} + (y - 5)^{2} = {3}^{2} \\ equating \: it \: with \: \\ (x - h)^{2} + (y - k)^{2} = {r}^{2} \\ - h = 7 \implies \: h = - 7 \\ - k = - 5 \implies \: k = 5 \\ center = ( - h, \: - k) = (7, \: \: - 5) \\ \\ radius \: r = 3](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%7By%7D%5E%7B2%7D%20%20%2B%2014x%20-%2010y%20%2B%2065%20%3D%200%20%5C%5C%20%20%7Bx%7D%5E%7B2%7D%20%20%2B%2014x%20%2B%20%20%7By%7D%5E%7B2%7D%20%20-%2010%20%2B%2065%20%3D%200%20%5C%5C%20%28%20%7Bx%7D%5E%7B2%7D%20%20%2B%2014x%20%2B%20%20%7B7%7D%5E%7B2%7D%29%20%20-%20%20%7B7%7D%5E%7B2%7D%20%20%2B%20%20%28%7By%7D%5E%7B2%7D%20%20-%2010y%20%2B%20%20%7B5%7D%5E%7B2%7D%29%20%20%5C%5C-%20%20%7B5%7D%5E%7B2%7D%20%20%2B%2065%20%3D%200%20%5C%5C%20%28x%20%2B%207%29%5E%7B2%7D%20%20%2B%20%28y%20-%205%29%5E%7B2%7D%20%20-%2049%20-%2025%20%2B%2065%20%3D%200%20%5C%5C%20%20%28x%20%2B%207%29%5E%7B2%7D%20%20%2B%20%28y%20-%205%29%5E%7B2%7D%20%20-%2049%20%20%2B%2040%3D%200%20%5C%5C%20%20%20%28x%20%2B%207%29%5E%7B2%7D%20%20%2B%20%28y%20-%205%29%5E%7B2%7D%20%20-%209%3D%200%20%5C%5C%20%20%20%28x%20%2B%207%29%5E%7B2%7D%20%20%2B%20%28y%20-%205%29%5E%7B2%7D%20%20%20%3D%20%209%20%5C%5C%20%20%28x%20%2B%207%29%5E%7B2%7D%20%20%2B%20%28y%20-%205%29%5E%7B2%7D%20%20%20%3D%20%20%20%7B3%7D%5E%7B2%7D%20%20%5C%5C%20%20equating%20%5C%3A%20it%20%5C%3A%20with%20%5C%3A%20%5C%5C%20%28x%20%20-%20%20h%29%5E%7B2%7D%20%20%2B%20%28y%20-%20k%29%5E%7B2%7D%20%20%20%3D%20%20%20%7Br%7D%5E%7B2%7D%20%20%5C%5C%20%20-%20h%20%3D%207%20%5Cimplies%20%5C%3A%20h%20%3D%20%20-%207%20%5C%5C%20%20-%20k%20%3D%20%20-%205%20%5Cimplies%20%5C%3A%20k%20%3D%205%20%5C%5C%20center%20%3D%20%28%20-%20h%2C%20%5C%3A%20%20-%20k%29%20%20%3D%20%287%2C%20%5C%3A%20%20%5C%3A%20%20-%205%29%20%5C%5C%20%20%5C%5C%20radius%20%5C%3A%20r%20%3D%203)
Answer:
[0,1] and [-1,0]
Step-by-step explanation:
We have the function
to determine consecutive values of x between which each real zero is located we have to replace the function with different values and then analyze the results.
We are going to replace the function with x=0,
![f(0)=-14.(0)^4-7.(0)^3-18.(0)^2+17.(0)+11\\f(0)=11](https://tex.z-dn.net/?f=f%280%29%3D-14.%280%29%5E4-7.%280%29%5E3-18.%280%29%5E2%2B17.%280%29%2B11%5C%5Cf%280%29%3D11)
Now we are going to replace with x=1,
![f(1)=-14.(1)^4-7.(1)^3-18.(1)^2+17.(1)+11\\f(1)=-14-7-18+17+11\\f(1)=-11](https://tex.z-dn.net/?f=f%281%29%3D-14.%281%29%5E4-7.%281%29%5E3-18.%281%29%5E2%2B17.%281%29%2B11%5C%5Cf%281%29%3D-14-7-18%2B17%2B11%5C%5Cf%281%29%3D-11)
With x=2,
![f(2)=-14.(2)^4-7.(2)^3-18.(2)^2+17.(2)+11\\f(2)=-224-56-72+34+11\\f(2)=-307](https://tex.z-dn.net/?f=f%282%29%3D-14.%282%29%5E4-7.%282%29%5E3-18.%282%29%5E2%2B17.%282%29%2B11%5C%5Cf%282%29%3D-224-56-72%2B34%2B11%5C%5Cf%282%29%3D-307)
Now with x=-1,
![f(-1)=-14.(-1)^4-7.(-1)^3-18.(-1)^2+17.(-1)+11\\f(-1)=-14+7-18-17+11\\f(-1)=-31](https://tex.z-dn.net/?f=f%28-1%29%3D-14.%28-1%29%5E4-7.%28-1%29%5E3-18.%28-1%29%5E2%2B17.%28-1%29%2B11%5C%5Cf%28-1%29%3D-14%2B7-18-17%2B11%5C%5Cf%28-1%29%3D-31)
With x=-2,
![f(-2)=-14.(-2)^4-7.(-2)^3-18.(-2)^2+17.(-2)+11\\f(-2)=-224+56-72-34+11\\f(-2)=-195](https://tex.z-dn.net/?f=f%28-2%29%3D-14.%28-2%29%5E4-7.%28-2%29%5E3-18.%28-2%29%5E2%2B17.%28-2%29%2B11%5C%5Cf%28-2%29%3D-224%2B56-72-34%2B11%5C%5Cf%28-2%29%3D-195)
We have to analyze:
We have that f(2) and f(1) have the same signs both are negatives this means that there isn't a zero between the interval [1,2].
We have that f(1) and f(0) have opposite signs this means that there is a zero between the interval [0,1].
f(0) and f(-1) have opposite signs too, then there's also a zero in the interval [-1,0].
And finally, f(-1) and f(-2) have the same signs then there isn't a zero in the interval [-2,-1]
The graph of the function shows that the answer is correct: