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3 years ago

What is the equation of a quadratic function p with rational coefficients that has a zero of 3-i

Mathematics

1 answer:

Kryger [21]3 years ago

6 0

Answer:

p = x² − 6x + 10

Step-by-step explanation:

Complex roots come in conjugate pairs. So if 3−i is a root, then 3+i is also a root.

p = (x − (3−i)) (x − (3+i))

p = x² − (3+i)x − (3−i)x + (3−i)(3+i)

p = x² − 3x − ix − 3x + ix + (9 − i²)

p = x² − 6x + 10

You can check your answer using the quadratic formula.

x = [ -b ± √(b² − 4ac) ] / 2a

x = [ 6 ± √(36 − 40) ] / 2

x = (6 ± 2i) / 2

x = 3 ± i

You might be interested in

Use completing the square to solve the equation x2 - 10x = -8.

Kay [80]

Answer:

x^2 - 10x + 25 = 17

Step-by-step explanation:

Please use " ^ " to denote exponentation:

x^2 - 10x = -8

x^2 - 10x + 25 = 17

x^2 - 10x + 25 = -8

x^2 - 10x - 25 = -33

Begin with x^2 - 10x = -8. Here's the process for completing the square:

1. Take half of the coefficient of x and square the result:

(1/2)(-10) = -5, and then (-5)^2 = + 25

2. Add this 25 to the first two terms of x^2 - 10x = -8 and also to the constant -8: x^2 - 10x + 25 = -8 + 25, or x^2 - 10x + 25 = 17

3. Match this result to one of the given possible answers. Turns out that the given possible answer matching our result is the first one:

x^2 - 10x + 25 = 17

7 0

3 years ago

Elton is a candle maker. Each 15 cm long candle he makes burns evenly for 6 hours if Elton makes a 45 cm long candle, how long w

tatuchka [14]

The candle would burn for 18 hours.

45 / 15 = 3

If the candle is three times longer than normal, it will burn three times longer than normal.

3 x 6 = 18

7 0

4 years ago

Calculate the average rate of change of f(x) over the interval [-4, -1] using the formula [ BLANK ] The value of f(-1) is [BLANK

lesantik [10]

Answers:
The formula is [f(-1)-f(-4)]/[3]
The value of f(-1) is 3
The value of f(-4) is -3
The average rate of change is 2

==============================================

Explanation:

For the first blank, we use the formula
[ f(b) - f(a) ]/[ b - a ]
where 'a' and 'b' are the endpoints for the x interval

In this case, a = -4 and b = -1. When you plug those values into the formula above, you get...
[ f(b) - f(a) ]/[ b - a]
[ f(-1) - f(-4)]/[ -1 - (-4) ]
[ f(-1) - f(-4)]/[ -1+4 ]
[ f(-1) - f(-4)]/[ 3 ]
which is why the answer is choice C for the first blank

-------------------------------------------

To compute the value of f(-1), we draw a vertical line through -1 on the x axis. This vertical line crosses the diagonal function graph at the point (-1,3). The y value of this point is what we want. Plugging in x = -1 leads to y = 3. This is why f(-1) = 3
If you want, you can draw a horizontal line through (-1,3) and you'll see it touching 3 on the y axis.

-------------------------------------------

Follow similar steps as above to compute f(-4). Draw a vertical line through x = -4 on the x axis. Mark the point where the vertical line crosses the diagonal line. This point is (-4,-3). Optionally draw a horizontal line over til you hit the y axis and you'll find that y = -3 corresponds to x = -4

This is why f(-4) = -3

-------------------------------------------

We'll use the last three sections to compute the average rate of change. Everything combines together building up to this moment.

From the first part, we had the formula
[ f(b) - f(a) ]/[ b - a ]
[ f(-1) - f(-4)]/[ 3 ]

We can replace the "f(-1)" with 3 since we found that f(-1) = 3
Similarly, f(-4) = -3 so we can replace the "f(-4)" with -3

Doing those replacements and simplifying leads to...

[ f(-1) - f(-4)]/[ 3 ]
[ 3 - (-3)]/[ 3 ]
[ 3 + 3]/[ 3 ]
6/3
2

So the average rate of change is 2

Note: because the entire graph is a straight line, the average rate of change for any interval a < x < b is going to be equal to the slope m. In this case, the slope of the line is m = 2/1 = 2. We move up 2 units each time we move to the right 1 unit along the diagonal line.

6 0

3 years ago

Write the equation of a line that is perpendicular to the line that passes through (4,−3)and (−7,8).

masha68 [24]

Line perpendicular to a given line is = 1/m

To find m, use m =y2-y1/x2-x1 formula

m = (8-(-3))/(-7-4)

m = 11/-11,

m = -1

Use y= mx+c to find equation of line

Plug in a pair of values,

-3= -1(4)+ c