What are the solutions to the equation?

1 answer:

8 0

So firstly, you want to set your equation to zero. You can do this by subtracting 40 on both sides of the equation. $x^2+6x-40=0$

Now, what two numbers add up to 6x and multiply to -40x^2? 10x and -4x. With this info, replace 6x with 10x and -4x: $x^2-4x+10x-40=0$

Next, factor x^2-4x and 10x-40 separately. Make sure that what's inside the parentheses is the same: $x(x-4)+10(x-4)=0$

Now you can rewrite the equation as: $(x+10)(x-4)=0$

Now using Zero Product Property, solve for x:

$x+10=0\\ x=-10$

$x-4=0\\ x=4$

In short, x = -10 and 4.

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VladimirAG [237]

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A gardener wants to make a rectangular enclosure using a wall as one side and 120 m of fencing for the other three sides. expres

Solnce55 [7]

The area is given by A = -2x² + 120x. The greatest area is given by x = 30 m.

Explanation:
See the picture attached for reference.

Let's call:
x = side not facing the wall
We know that the total fence is 120m, therefore
(120 - 2x) = side facing the wall

Note: you could choose to be x = side facing the wall, but the calculations would be a little bit more complicated.

We can now calculate the area of a rectangle:
A = b · h =
= x · (120 - 2x)
= -2x² + 120x

In order to find a maximum for this function, we need to calculate the first derivative:
$\frac{d}{dx} (-2x^{2} + 120x ) = -4x +120$

Then, we need to find the candidate points by setting the derivative equal to zero:
-4x +120 = 0
-4(x - 30) = 0
x = 30

Now, in order to understand if the candidate point is a maximum or a minium, let's calculate the second derivative:

 $\frac{d}{dx}(-4x + 120) = -4$ 

According to the "Second Derivative Test", if the second derivative is negative, the point is a local maximum.

Hence, x = 30m gives the greatest area, and we would have:
side not facing the wall = 30m
side facing the wall = 120 - 2·30 = 60m
area = 30 · 60 = 1800 m²