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3 years ago

Solve the equation by completing the square. Round to the nearest hundredth if necessary x^2+3x=25

Mathematics

2 answers:

Citrus2011 [14]3 years ago

7 0

Answer:

The solutions are: $x= 3.72, -6.72$

Step-by-step explanation:

$x^2+ 3x = 25\\ \\ x^2+3x+(\frac{3}{2})^2 = 25 +(\frac{3}{2})^2\\ \\ (x+\frac{3}{2})^2 = 25+\frac{9}{4}\\ \\ (x+\frac{3}{2})^2 =27.25\\ \\ \sqrt{(x+\frac{3}{2})^2} =\pm \sqrt{27.25} \\ \\ x+1.5= \pm \sqrt{27.25} \\ \\ x= -1.5 \pm \sqrt{27.25}\\ \\ x=-1.5+ \sqrt{27.25}=3.72015... \approx 3.72\\ \\ or\\ \\ x=-1.5-\sqrt{27.25} =-6.72015...\approx -6.72$

So, the solutions are: $x= 3.72, -6.72$

Lilit [14]3 years ago

6 0

X=-3/2 plus or minus to the square root 109/2

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Dale works at a mattress store, and makes a salary plus commission. He has a $500 weekly salary and makes a 5% commission for sa

Anni [7]

Answer:

1. $\( f\circ g(x)=0.05x-150$

2. $\( g\circ f(x)=0.05x-3000$

3. The first one represents Dale's commission

Explanation:

1. The composition of the function

$f\circ g(x)=f(g(x))$

means that you first apply the function g(x) and then f(x) on the output of g(x).

That is:

f(x) = 0.05x
g(x) = x - 3000

$f(g(x)=0.05(x - 3000)$

$f(g(x))=0.05x-150$

2. The composition of the function

$g\circ f(x)=g(f(x))$

means that you first apply the function f(x) and then g(x) on the output of f(x).

That is:

$g(f(x))=((0.05x)-3000)=0.05x-3000$

3. Which one represents Dale's commission

To calculate Dales's commision you must subtract $3,000 from the sales, to find the sales over $3000. That is: x - 3,000, which is the function g(x).

Therefore, you first use g(x).

Then, you must multiply the output of g(x) by 0.05 to find the 5% of the sales over $3,000. That is: 0.05(g((x)) = 0.05(x - 3000) = 0.05x - 150.

Therefore, the composition that represents Dale's commission is the first one:

$f(g(x)=0.05(x - 3000)$

$f(g(x))=0.05x-150$

4 0

3 years ago

Let z=3+i, then find a. Z² b. |Z| c.<img src="https://tex.z-dn.net/?f=%5Csqrt%7BZ%7D" id="TexFormula1" title="\sq

zysi [14]

Given z = 3 + i, right away we can find

(a) square

z ² = (3 + i )² = 3² + 6i + i ² = 9 + 6i - 1 = 8 + 6i

(b) modulus

|z| = √(3² + 1²) = √(9 + 1) = √10

(d) polar form

First find the argument:

arg(z) = arctan(1/3)

Then

z = |z| exp(i arg(z))

z = √10 exp(i arctan(1/3))

z = √10 (cos(arctan(1/3)) + i sin(arctan(1/3))

Any complex number has 2 square roots. Using the polar form from part (d), we have

√z = √(√10) exp(i arctan(1/3) / 2)

and

√z = √(√10) exp(i (arctan(1/3) + 2π) / 2)

Then in standard rectangular form, we have

$\sqrt z = \sqrt[4]{10} \left(\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) + i \sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right)\right)$

and

$\sqrt z = \sqrt[4]{10} \left(\cos\left(\dfrac12 \arctan\left(\dfrac13\right) + \pi\right) + i \sin\left(\dfrac12 \arctan\left(\dfrac13\right) + \pi\right)\right)$

We can simplify this further. We know that z lies in the first quadrant, so

0 < arg(z) = arctan(1/3) < π/2

which means

0 < 1/2 arctan(1/3) < π/4

Then both cos(1/2 arctan(1/3)) and sin(1/2 arctan(1/3)) are positive. Using the half-angle identity, we then have

$\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1+\cos\left(\arctan\left(\dfrac13\right)\right)}2}$

$\sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1-\cos\left(\arctan\left(\dfrac13\right)\right)}2}$

and since cos(x + π) = -cos(x) and sin(x + π) = -sin(x),

$\cos\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{1+\cos\left(\arctan\left(\dfrac13\right)\right)}2}$

$\sin\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{1-\cos\left(\arctan\left(\dfrac13\right)\right)}2}$

Now, arctan(1/3) is an angle y such that tan(y) = 1/3. In a right triangle satisfying this relation, we would see that cos(y) = 3/√10 and sin(y) = 1/√10. Then

$\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1+\dfrac3{\sqrt{10}}}2} = \sqrt{\dfrac{10+3\sqrt{10}}{20}}$

$\sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1-\dfrac3{\sqrt{10}}}2} = \sqrt{\dfrac{10-3\sqrt{10}}{20}}$

$\cos\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{10-3\sqrt{10}}{20}}$

$\sin\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{10-3\sqrt{10}}{20}}$

So the two square roots of z are

$\boxed{\sqrt z = \sqrt[4]{10} \left(\sqrt{\dfrac{10+3\sqrt{10}}{20}} + i \sqrt{\dfrac{10-3\sqrt{10}}{20}}\right)}$

and

$\boxed{\sqrt z = -\sqrt[4]{10} \left(\sqrt{\dfrac{10+3\sqrt{10}}{20}} + i \sqrt{\dfrac{10-3\sqrt{10}}{20}}\right)}$

3 0

3 years ago

Read 2 more answers

The sum of 3 consecutive integers is 45. Which equation represents this?

cestrela7 [59]

Consecutive integers are 1 apart
they are
x,x+1,x+2
so it would be
x+x+1+x+2=45
B is answer

3 0

3 years ago

What is the value of a?

kaheart [24]

Answer:

$a=\dfrac{16}{3}\ un.$

Step-by-step explanation:

In right triangle XZW, the height WY is the geometric mean of segments ZY and XY.

Hence,

$WY^2 =ZY\cdot XY,\\ \\4^2=3\cdot a,\\ \\a=\dfrac{16}{3}\ un.$

7 0

3 years ago

Read 2 more answers

-30 + 10 Find the sum

ludmilkaskok [199]

The answer of -30+10 is -20

-30+10=-20

5 0

3 years ago