Roots and Radicals
root(5)(n^20)
n^4
(-2-5i)(-4+3i)
23+14i
(2x+7)^2=25
x=-6, x=1
Complete the Square
x^2+12x=-20
x=-10, x=-2
Identify the vertex and two additional points (one to each side of the vertex)
f(x)=x^2
Vertex (0,0)
(2,4), (-2,4), (3,9) etc.
Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.
The base is 10in and the height is 14in
((27b^(2/3)c^(-5/2))/(b^(-7/3)c^(1/2)))^(1/3)
(3b)/c
(6+i)-(-2-4i)
8+5i
(n-4)^2-50=150
n=4+-10sqrt(2)
5z(z-2)=3
z=(5+-2sqrt(10))/5
Find the vertex and x-intercepts.
f(x)=x^2-x+2
Vertex
(1/2, 7/4)
X-intercepts (-1,0), (2,0)
A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula
h=-16t^2+v_0t
to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.
t=4 (t=6)
sqrt(5)/(sqrt(n)-sqrt(7)
(sqrt(5n)-sqrt(35))/(n-7)
Rationalize
-(4)/(3-2i)
-12/13-8/13i
(z-10)(z+2)=28
z=-4, z=12
9d^2-12d=-4
d=2/3
x^2+2x>=24
x in (-infty, -6)cup (4, infty)
A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation h=-16t^2+112t to find how long it will take the ball to reach maximum height, and then find the maximum height.
t=3.5
h_(max)=196
(8x+5)^(1/3)+2
x=-4
(1+sqrt(-36))(3-sqrt(-25))
33-13i
x^6-19x^3-216=0
x=-2, x=3
x^(2/3)-10x^(1/3)+24=0
x=64, x=216
x^2>0
x in (-infty,0)cup(0,infty)
A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation A(x)=x(90-x/2)
gives the area, A, of the yard for the length, x, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.
The length adjacent to the building is 90 feet giving a maximum area of 4,050 square feet.
7root(3)(24x^3)-12root(3)(24x^4)+xsqrt(24x^4)
-10xroot(3)(3x)+4x^3sqrt(6)
(sqrt(4)+sqrt(-9))^2+(sqrt(12)-sqrt(-289))
-5+2sqrt(3)-5i
8x^(-2)-2x^(-1)-3=0
x=-2, x=4/3
(sqrtx-2)^2-3(sqrtx-2)-15=0
x=49, x=0
(x^2-1)/x<=0
x in (-infty,1]cup(0,1]
Ezra kayaked up the river and then back in a total time of 6 hours. The trip was 4 miles each way and the current was difficult. If Roy kayaked at a speed of 5 mph, what was the speed of the current? Round to the nearest hundreth.
3.53mph