Solve 5x2 - 7x+2 -0 by completing the square. What are the solutions?

I read this as -5e⁻⁴ˣ⁺²+3=½log₁₀(x²+1).

If this is not right the steps below can guide you to a possible solution when the correct formula is used.

My calculator is a Casio fx-85GT PLUS which has a function facility (MODE 3: TABLE).

I input the function: -5e⁻⁴ˣ⁺²+3-½log₁₀(x²+1). I assumed that log was log to base 10 rather than natural log ln.

The facility asks me to input a start value, end value and step value. I chose -10, 10 and 1 initially.

After a few seconds of evaluating a table of results the calculator showed me the results of plugging in x=-10 up to 10 in steps of 1.

The results showed the evaluation for each of the 21 values and I looked for the sign of the result to change.

For x=0 the result was -33.95 approx and for x=1 it was 2.1728 approx. So between 0 and 1 there is a zero, a solution for x.

I returned to the function and changed the parameters to start=0, end=1, step=0.1 to begin the next evaluation.

This time the sign change occurred between x=0.6 and 0.7.

I returned to the function with parameters: start=0.6, end=0.7, step=0.01.

The sign change occurred between 0.63 and 0.64, the start and end parameters with step=0.001 for the next iteration.

The solution is between 0.633 and 0.634.

Repeat the process one more time with step=0.0001. The sign change was between 0.6338 and 0.6339 so the solution to three dec places is 0.634.

Step-by-step explanation:

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).