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3 years ago

show that thw roots of the equation (x-a)(x_b)=k^2 are always real if a,b and k are real. Please I really need help with this

Mathematics

1 answer:

VLD [36.1K]3 years ago

8 0

Answer:

see explanation

Step-by-step explanation:

Check the value of the discriminant

Δ = b² - 4ac

• If b² - 4ac > 0 then roots are real

• If b² - 4ac = 0 roots are real and equal

• If b² - 4ac < 0 then roots are not real

given (x - a)(x - b) = k² ( expand factors )

x² - bx - ax - k² = 0 ( in standard form )

x² + x(- a - b) - k² = 0

with a = 1, b = (- a - b), c = -k²

b² - 4ac = (- a - b)² + 4k²

For a, b, k ∈ R then (- a - b)² ≥ 0 and 4k² ≥ 0

Hence roots of the equation are always real for a, b, k ∈ R

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This question is incomplete and it lacks an attached graph. Find attached to this answer, the appropriate graph.

Complete Question

Carmen and Matt are conducting different chemistry experiments in school. In both experiments, each student starts with an initial amount of water in a flask. They combine two chemicals which react to produce more water. Carmen's experiment starts with 30 milliliters of water in a flask, and the water increases in volume by 8.5 milliliters per second. Matt's experiment starts with 10 milliliters of water and increases in volume by 28% each second. The graph represents the volume of water in the two flasks in relation to time. Which two conclusions can be made if f represents the volume of water in Carmen's flask and g represents the volume of water in Matt's flask? a) The volume of water in Carmen's flask is increasing at a slower rate than the volume of water in Matt's flask over the interval [0, 2].

b) The volume of water in Carmen's flask is increasing at a faster rate than the volume of water in Matt's flask over the interval [6, 8].

c) The volume of water in Carmen's flask will always be greater than the volume of water in Matt's flask.

d) The volume of water in Matt's flask will eventually be greater than the volume of water in Carmen's flask.

e) The volume of water in Carmen's flask is increasing at a slower rate than the volume of water in Matt's flask over the interval [4, 6].

Answer:

c) The volume of water in Carmen's flask will always be greater than the volume of water in Matt's flask.

e) The volume of water in Carmen's flask is increasing at a slower rate than the volume of water in Matt's flask over the interval [4, 6].

Step-by-step explanation:

Looking at the graph, the average rate of change is given as [0,2], [4,6], and [6,8].

From the graph and question, we can tell that Matt's experiment starts with 10 milliliters of water and increases in volume by 28% each second

Which would gives us:

28% of 10 milliliter of water = 2.8 millimeters per second.

For Carmen , his experiment starts with 30 milliliters of water in a flask, and the water increases in volume by 8.5 milliliters per second,

Because, Carmen starts with 30 milliliters of water in his flask, the volume of water in Carmen's flask will always be greater than the volume of water in Matt's flask.

Secondly, comparing the quantity as which the volume of the flask increases per second, at the interval of [4, 6], the volume of water in Carmen's flask is increasing at a slower rate than the volume of water in Matt's flask over the interval

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