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3 years ago

How do i solve functions?

Mathematics

1 answer:

Maksim231197 [3]3 years ago

6 0

Most quadratic functions(which is what you have there, to a degree of 2) are solved using factoring and the zero product law. If you can not factor then you have to use the quadratic formula or graph it. However this one can be factored.
It's pretty simple to just factor it by inspection but I use the chart method, if you know decomposition that works as well.
Factoring gives us,
$(2x + 1)(x - 3) = 0$
Then you set each factor to 0 and solve for x,
$2x + 1 = 0$
$2x = - 1$
$x = \frac{ - 1}{2}$
And the second one,
$x - 3 = 0$
$x = 3$
The solutions to this equation are
x = -1/2, 3

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Let the (x; y) coordinates represent locations on the ground. The height h of

grigory [225]

The critical points of h(x,y) occur wherever its partial derivatives $h_x$ and $h_y$ vanish simultaneously. We have

$h_x = 8-4y-8x = 0 \implies y=2-2x \\\\ h_y = 10-4x-12y^2 = 0 \implies 2x+6y^2=5$

Substitute y in the second equation and solve for x, then for y :

$2x+6(2-2x)^2=5 \\\\ 24x^2-46x+19=0 \\\\ \implies x=\dfrac{23\pm\sqrt{73}}{24}\text{ and }y=\dfrac{1\mp\sqrt{73}}{12}$

This is to say there are two critical points,

$(x,y)=\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)\text{ and }(x,y)=\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)$

To classify these critical points, we carry out the second partial derivative test. h(x,y) has Hessian

$H(x,y) = \begin{bmatrix}h_{xx}&h_{xy}\\h_{yx}&h_{yy}\end{bmatrix} = \begin{bmatrix}-8&-4\\-4&-24y\end{bmatrix}$

whose determinant is $192y-16$ . Now,

• if the Hessian determinant is negative at a given critical point, then you have a saddle point

• if both the determinant and $h_{xx}$ are positive at the point, then it's a local minimum

• if the determinant is positive and $h_{xx}$ is negative, then it's a local maximum

• otherwise the test fails

We have

$\det\left(H\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)\right) = -16\sqrt{73} < 0$

while

$\det\left(H\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)\right) = 16\sqrt{73}>0 \\\\ \text{ and } \\\\ h_{xx}\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)=-8 < 0$

So, we end up with

$h\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)=-\dfrac{4247+37\sqrt{73}}{72} \text{ (saddle point)}\\\\\text{ and }\\\\h\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)=-\dfrac{4247-37\sqrt{73}}{72} \text{ (local max)}$

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3 years ago

Gus has skydived 4 more times than Nico. Emma has skydived twice as many times as Nico. If Emma has skydived 16 times, how many

Oliga [24]

32 times bc emma skydived 16 times and it says sue sky dived 2 × Nico so you would divide 16 ÷ 2 and get 8 then it says gus has skydived 4 times more than nico so you do 8 × 4 and get 32 as your answer

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4 years ago

A catfish pond contains 2,200 gallons of water. The pond can hold no more than 3,000 gallons. It is being filled at a rate of 12

Olenka [21]

Answer:

I am not sure honestly, but I think you should

Step-by-step explanation:

add 2,200 and 3,000 then see your answer, and divide it by 125. I'm sorry if this doesn't help......

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3 years ago

Which equation represents the graphed function?

Ilia_Sergeevich [38]

Answer:

y = -2x + 3

Step-by-step explanation:

If you use the formula to find slope, which is y2-y1 over x2-x1, you can find the slope is -2. the only equation that has -2 as the slope is the first one, y = -2x + 3. hope I helped!

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3 years ago

An emerald pendant was marked up 150% from an original cost of $488. Last Friday, Cameron bought the emerald pendant and paid an

Marysya12 [62]

Given parameters:

Mark up percentage = 150%

Original cost = $488

Sales tax = 5%

Unknown:

His total cost = ?

Solution:

Marking up a good is the amount a sell adds to a product which makes its selling price usually for a profit.

Since the emerald was marked up by 150%;

New cost = (1 + 1.5) x $488 = $1220

Now, the additional tax was 5%;

Tax = $\frac{5}{100}$ x 1220 = $61

So, the total cost = New cost + tax = $1220 + $61 = $1281

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3 years ago