1.
The x-coordinate of the vertex is b/2, where b is the variable in ax2 +bx +c, when a < 0
So, in your problem b/2 = -4/2 = -2
Now plug in –2 for x to get y, the y-coordinate of the vertex
Y = -(-2)2 – 4(-2) +5 = -4 +8 +5 = 9
The vertex is (-2, 9)
Alternatively, you could complete the square – have you learned this???it can be done so many different
ways – here is how I do it
y = -x2 – 4x + 5
-y = x2 + 4x – 5
-y + 5 = x2 + 4x, take (4/2)2 = 4 and add to both sides
-y + 5 + 4 = x2 + 4x + 4, now factor the right hand side
-y + 5 + 4 = (x +2)2, hence the name “completing the square”…now solve for y
-y + 9 = (x +2)2
-y = (x +2)2 – 9
y = -(x +2)2 + 9, now this gives the vertex (h,k) in y = a(x – h)2 + k, so ours is (-2, 9)
2.
4x2 – 20x – 24 = 4[x2 – 5x – 6] = 4(x-6)(x+1), you can always check by multiplying
3.
12x2 – 75 = 3[4x2 – 25], which is now a difference of squares (2x)2 - 52 = (2x - 5)(2x + 5)
answer 3(2x – 5)(2x + 5), you can always check by multiplying
4.
8y3 – 27 = (2y)3 - 33, this is a simple difference of cubes
=(2y – 3)[(2y)2 +3(2y) + 32]
=(2y – 3)(4y2 +6y + 9), you can always check by multiplying
5.
2x3y – 12x2y – 14xy = 2xy[x2 – 6x – 7] = 2xy(x-7)(x+1), you can always check by multiplying
6.
4x2 – 4xy + y2 = (2x – y)(2x – y) = (2x – y)2 … The binomial theorem, you can check by multiplying
7.
(6x – 5y)(2x + 3y), found by trial & error…you can check by multiplying
8.
If you know imaginary numbers, this is a simple sum of squares rule…
answer (x-6i)(x + 6i)
If you don’t know what I mean by imaginary numbers, then your answer is that the problem cannot
be factored
9.
(2x + 3)(3x - 4)… found by trial & error…you can check by multiplying
10.
the LCM of all denominators is 3(y + 2), so multiply the whole equation by it and you get
3(6) –4(3)(y + 2) = y + 20
18 –12(y + 2) = y + 20
18 – 12y – 24 = y + 20
-6 – 12y = y + 20
-26 = 13y
-2 = y, now you must check this since the original problem had a denominator… and in the original
problem –2 is not in the domain because it would make a denominator zero…so –2 is not a solution… so
our answer is no solution
11.
the LCM of all denominators is 4(x + 3), so multiply the whole equation by it and you get
5 + (x + 3)(x – 3) = 5
(x + 3)(x – 3) = 0
x2 – 9 = 0
x2 = 9
x = ± 3, but as in problem #10, you cannot have a zero in the denominator and –3 makes a zero in the
denominator of the original problem so throw out x = -3 and your solution is x=3
12.
(x – 7)2 = x – 5, by squaring both sides of the original problem
x2 – 14x + 49 = x –5
x2 – 15x + 54 = 0
(x – 6)(x – 9) = 0, by factoring the left hand side
x – 6 = 0 and x – 9 = 0
x = 6 and x = 9…check these answers and you find that x = 6 doesn’t work in original problem so throw it
away
your final answer is x = 9
13.
√(2x – 4) = -1, no solution because a square root is always positive
14.
x2 + 6x = 5
x2 + 6x – 5 = 0, cannot quickly be factored so use the quadratic equation with A = 1 B=6 C= -5
[-6 ± √ (62 – 4(1)(-5)) ] / [2(1)] = [-6 ± √ (36 +20) ] / 2 = [-6 ± √ 56 ] / 2 = [-6 ± 2√ 14 ] / 2
= -3 ± √ 14
your 2 answers are x= -3 + √ 14 or x= -3 - √ 14
15.
(3x + 5)(4x – 2) = 10
12x2 – 6x + 20x – 10 = 10
12x2 + 14x – 10 = 10
12x2 + 14x – 10 – 10 = 0
12x2 + 14x – 20 = 0
2[6x2 + 7x – 10] = 0
6x2 + 7x – 10 = 0 , by dividing both sides by 2
(6x – 5)(x + 2) = 0, by factoring the left hand side
6x – 5 = 0 or x + 2 = 0
x = 5/6 or x = -2
16.
2x2 – 6x – x + 3 = 7, by multiplying out the left hand side of the original problem
2x2 – 6x – x + 3 - 7 = 0
2x2 – 7x - 4 = 0
(2x + 1)(x – 4) = 0, by factoring the left hand side
2x + 1 = 0 or x – 4 = 0
x = -1/2 or x = 4
17.
321 = -16t2 – 2t + 524
16t2 + 2t – 524 + 321 = 0
16t2 + 2t – 203 = 0
Use Quadratic Equation with A= 16 B=2 C=-203…
[-2 ± √ (22 – 4(16)(-203)) ] / [2(16)] = [-2 ± √ (4 + 12992) ] / [2(16)] = [-2 ± √12996 ] / [2(16)]
= [-2 ± 114 ] / [2(16)] = [-1 ± 57 ] / 16
so your 2 answers are t = (-1 + 57) /16 = 56/16 = 7/2 = 3.5
and t = (-1 - 57) /16 = -58/16 = -29/8, but time cannot be negative
so throw this one out
Your final answer is: the rock is 321 ft from the ground at 3.5 seconds
18.
plug 72 in for v and solve for L
72 = 2√6L
72/2 = 2√6L /2
36 = √6L
362 = 6L
1296/6 = L
216 = L
Answer: at a speed of 72 miles per hour, the car will skid 216 feet.
19.
coefficient of x2 term is positive so it opens up. Leaving the bottom 2 graphs as options
If x = 0 then y = 4 so the y-intercept is 4 so it looks like the bottom left graph
If x = 1 then y = 0, so one of the x-int is x=1 so it is the bottom left graph for sure
20.
coefficient of x2 term is negative so it opens down. Eliminating the bottom right graph
if x = 0 then y = 3 so the y-intercept is 3…giving us the top right graph as the answer
one more point to check: if x = 1 then y = 4, so answer is the top right graph