Math Help - Need help on these 3 problems

1. Need help on these 3 problems

I really need help with these.

Solve the equation for x. (If there is one solution with a multiplicity of two, enter it in both answer boxes.)
a^2x^2 + 8ax + 16 = 0
i think there is a larger and smaller value

Find all values of k that ensure that the given equation has exactly one solution.
6x^2 + kx + 9 = 0

there is a smaller and larger value

A wire 370 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire?
No idea on this one =(

2. Originally Posted by Fuze
I really need help with these.

Solve the equation for x. (If there is one solution with a multiplicity of two, enter it in both answer boxes.)
a^2x^2 + 8ax + 16 = 0
i think there is a larger and smaller value

Find all values of k that ensure that the given equation has exactly one solution.
6x^2 + kx + 9 = 0
a quadratic has one root if it's discriminant is zero.

the discriminant for a quadratic of the form $y = ax^2 + bx + c$ is given by $\triangle = b^2 - 4ac$

A wire 370 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire?
No idea on this one =(
Let $x$ be the length that is cut to make the square. then the length to make the circle will be $370 - x$ (this will be the circumference of the circle).

the side-length of the square will be $\frac x4$ and we can use the formula for the circumference of a circle to find the radius of a circle whose circumference is $370 - x$. thus we can find formulas for the areas of the two figures. now we just equate them and solve for x

(if you wish, you can let the length for the square be 4x, this avoids the awkwardness of x/4, but you should note that when you solve for x, you must multiply the value by 4 again to get the answer for the length of this piece)

3. Originally Posted by Fuze
I really need help with these.

Solve the equation for x. (If there is one solution with a multiplicity of two, enter it in both answer boxes.)
a^2x^2 + 8ax + 16 = 0
i think there is a larger and smaller value

Originally Posted by Fuze

Find all values of k that ensure that the given equation has exactly one solution.
6x^2 + kx + 9 = 0

there is a smaller and larger value
Delta = 0

$b^2 - 4ac = 0$

$k^2 - 4(6)(9) = 0$

$k^2 = 216$

$k = 6 \sqrt{6}$

Originally Posted by Fuze
A wire 370 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire?
No idea on this one =(
One wire length = $x$
The other = $370 - x$

Perimeter of square = $4L$

Area of square = $L^2$

So $4L = 370 - x$

Then $L = \frac{370 - x}{4}$

$L^2 = \frac{(370 - x)^2}{16}$

Perimeter of circle = $2 \pi r$
Area of a circle = $\pi r^2$

If $2 \pi r = x$ then:

$r = \frac{x}{2 \pi}$

Area of the circle $= \pi \left( \frac{x^2}{4 (\pi)^2} \right)$

If the area is the same then $L^2 = \pi r^2$

And you can take it from here.

4. Correction

On the second problem:
There are actually two different values for k that solve the problem:
$k=\pm6\sqrt{6}$

--Kevin C.

5. Originally Posted by TwistedOne151
On the second problem:
There are actually two different values for k that solve the problem:
$k=\pm6\sqrt{6}$

--Kevin C.
Indeed! don't i look stupid!

thanks for the catch Kevin

6. thanks guys! really helped me alot