00:01
If the length of a rectangle is x plus 2, its area is x squared minus 10x minus 24, what is the other dimension? so we have a rectangle whose length is x plus 2, and the area is equal to x squared minus 10x minus 24.
00:27
Area is equal to length times width.
00:36
We know the length as x plus two.
00:40
So we need to find the width.
00:42
So we're going to reverse foil and we'll find that x is going to be equal to minus 12 on the right hand side.
00:51
So therefore, the width of the triangle, or excuse me, the rectangle is equal to x minus 12.
01:04
Question two.
01:07
The sum of two numbers, x plus y, is equal to 11.
01:13
The product of those two numbers is equal to 30.
01:19
What are the two numbers? so we can rearrange this first equation to equal y equals 11 minus x.
01:33
Then substitute that into the second one.
01:38
And that will be x times 11 minus x equals.
01:43
30.
01:44
If we distribute the x's, we'll have 11x minus x squared is equal to 30.
01:51
I'm going to move everything over to the other side.
01:54
So we'll have 0 is equal to x squared minus 11x plus 30.
02:03
Now we just have to reverse foil.
02:06
So we'll have x and x on the left of the parentheses.
02:10
And we need two numbers that when multiplied will equal 30 and when summed will equal 11.
02:19
So that will be minus 6 and minus 5.
02:26
So the numbers are going to be 5 and 6.
02:34
Question 3.
02:37
Suppose that thrice the square of a number is 12 times that number.
02:42
What are the possible numbers? so three times a number squared is equal to 12 times that number.
02:55
So we have 3x squared minus 12x is equal to 0.
03:02
We can factor out an x and we will get 3x minus 12 is equal to 0.
03:10
So x can be 12 over 3 or 0.
03:16
Question four.
03:19
The area of a square is numerically equal to four times its perimeter.
03:25
What is the measurement of a side of the square? the area of a square is numerically equal to four times its perimeter.
03:33
What is the measure of the side of a square? so we have a square, and the area of that square is equal to four times the perimeter.
03:46
If we have a square, and the area of that square is equal to four times the perimeter.
03:47
Side 1, 2, 3, and 4, the area is equal to length times width.
03:59
Or we can say side 1 times side.
04:07
The perimeter is equal to 1 plus 2 plus 3 plus.
04:15
Since it's a square, we know that 1 equals 4 and 2 equals 3.
04:21
So all of the sides are equal.
04:24
So we can say the area is equal to x squared, while the perimeter is equal to 4 times x.
04:41
So we'll say x squared is equal to 4x.
04:50
We'll move the 4x over and factor out 1x, so then we'll have x is equal to x minus 4, have that equal to 0.
05:01
X therefore can equal 0 or positive 4.
05:08
Since you can't have a side of 0, x must be.
05:14
So the side of the square is four minutes.
05:20
Question 5.
05:24
40 less than the square of a number is equal to six times that number.
05:35
So let's rewrite that.
05:36
So we have x squared minus 40 is equal...