1. Expand the product $(t-2)(4t^2 + 16)(t+2)$.
2. Find all values of $x$ that satisfy the equation \[ \frac {12x}{x^2 + 8} = 2. \]
3. Compute the sum of all the solutions of $(3x+1)(x-7)+(x-3)(3x+1)=0$. Express your answer as a fraction.
4. Let $a$ and $b$ be the solutions of the quadratic equation $2x^2 - 8x + 7 = 0$. Find \[\frac{1}{2a} + \frac{1}{2b}.\]
5. Let $s$ and $t$ be the solutions of the quadratic $4x^2 + 9x - 6 = 0.$ Find $$\frac st + \frac ts.$$
+128631 1. Expand the product \($(t-2)(4t^2 + 16)(t+2)$\)
Rewrite as ( t - 2) (t + 2) (4t^2 + 16) =
(t^2 - 4) * 4 * (t^2 + 4) =
4* (t^2 - 4) (t^2 + 4) =
4 (t^4 - 16) =
4t^4 - 64
2. Find all values of x that satisfy the equation \(\frac {12x}{x^2 + 8} = 2\)
Multiply both sides by x^2 + 8
12x = 2 (x^2 + 8) divide both sides by 2
6x = x^2 + 8 subtract 6x from both sides and rearrange
x^2 - 6x + 8 = 0 factor
(x - 4) (x - 2) = 0
Setting both factors to 0 and solve for x and we get that
x = 4 and x = 2
3. Compute the sum of all the solutions of \( $(3x+1)(x-7)+(x-3)(3x+1)=0$\) . Express your answer as a fraction.
We can factor this as (3x + 1) ( x - 7 + x - 3) = 0 simplify
(3x + 1) (2x - 10) = 0
(3x +1) * 2 * (x - 5) = 0 divide both sides by 2
(3x + 1)(x - 5) = 0
Setting each factor to 0 and solving for x gives that x = -1/3 and x = 5
The sum of these is 5 - 1/3 = 15/3 - 1/3 = 14 / 3 = 4 + 2/3
EDIT to correct a small typo....thanks to the guest for spotting my error !!
CPhill: I get a slightly different answer on No. 3 as follows:
Solve for x:
(x - 7) (3 x + 1) + (x - 3) (3 x + 1) = 0
Expand out terms of the left hand side:
6 x^2 - 28 x - 10 = 0
The left hand side factors into a product with three terms:
2 (x - 5) (3 x + 1) = 0
Divide both sides by 2:
(x - 5) (3 x + 1) = 0
Split into two equations:
x - 5 = 0 or 3 x + 1 = 0
Add 5 to both sides:
x = 5 or 3 x + 1 = 0
Subtract 1 from both sides:
x = 5 or 3 x = -1
Divide both sides by 3:
x = 5 or x = -1/3. So, the sum is: 5 - 1/3 =4 2/3
+128631 4. Let a and b be the solutions of the quadratic equation 2x^2 - 8x + 7 = 0
Find 1 / [2a] + 1 / [2b] = (1/2) (1/a + 1/b) = (1/2) (a + b) / [ab] =
[ a + b ] / [2ab]
This isn't as hard as it seems
The sum of the roots = - [-8 ] / 2 = 4
So a + b = 4
The product of the roots = 7/2
So ab = 7/2 ⇒ 2ab = 7
So
[ a + b ] / [ 2ab] = 4 / 7
+128631 5. Let s and t be the solutions of the quadratic 4x^2 + 9x - 6 = 0. Find
s + t s^2 + t^2
__ ___ = ________
t s st
This is much like 4 with a few twists
The sum of the roots = -9/4
So s + t = -9/4
Square both sides
s^2 + 2st + t^2 = 81/16 (1)
The product of the roots = -6/4 = -3/2
So st = -3/2 ⇒ 2st = - 3 (2)
Sub (2) into (1) and we have that
s^2 - 3 + t^2 = 81/16 add 3 to both sides
s^2 + t^2 = 81/16 + 3
s^2 + t^2 = 81/16 + 48/16
s^2 + t^2 = 129/16
So
s^2 + t^2 (129/16) - (129/16) (2/3) = - (129/3) (2/16) =
_________ = _________ =
st - (3/2)
- 43 ( 1/ 8) =
-43 / 8
