A rectangular kitchen that is 8 feet longer than it is wide with width w would have length l = w + 8, or w = l − 8. Since the area A = 240 = lw, substitute w = l − 8 into 240 = lw to get 240 = l(l − 8) = l² − 8l, or l² −8l−240 = 0. To find l, factor l²−8l−240 = 0 into (l + 12)(l − 20) = 0. Thus l = −12 (which doesn’t make sense because a kitchen cannot have a negative length), or l = 20.

A circle with center (h, k) and radius r has equation (x−h)²+(y−k)² = r². In order to determine the equation of the circle in the standard (x, y) coordinate plane that has a radius of 4 units and the same center as the circle determined by x² + y² − 6y + 4 = 0, first determine the center of the circle defined by x² + y² − 6y + 4 = 0 by converting the equation to the standard form for a circle (complete the square):
x² + y² − 6y + 4 = 0
x² + y² − 6y + 9 + 4 − 9 = 0
(no net change in value)
x² + (y² − 6y + 9) + 4 − 9 = 0 (regroup)
x² + (y − 3)2 + −5 = 0 (factoring)
x² + (y − 3)² = 5
Thus the center of the circle is (0,3). A circle that has a radius of 4 units and the same center has the equation x² + (y − 3)2 = 16.

If the perimeter of a square is 36 units, then each side is 9 (since perimeter in a square is 4(s) where s is the length of a side). To find the length of the diagonal, you can use the Pythagorean Theorem because the diagonal is the hypotenuse of a right triangle with legs of length 9.
Thus d² = 9² +9², or 81+81, which equals 2(81),
and d = \(\sqrt{(2(81))}\), or √2 \(\sqrt{81}\), which is 9√2.

To solve this problem, remember that |3a − 3| ≥ 12 is equivalent to 3a − 3 ≥ 12 or 3a − 3 ≤ −12. Adding 3 to both sides and dividing by 3 yields a ≥ 5 or a ≤ −3.

Any integer when multiplied by an even integer results in an even integer. Also, the addition of any two even integers yields another even integer. Thus, regardless of the value of a or the power of a, multiplying by an even integer and adding an even integer will yield an even integer. This only occurs in the calculation 6a⁴ + 6.

Systems of equations have an infinite number of solutions when the equations are equivalent (i.e. they graph the same lane). In order for the two equations to be equivalent, the constants and coefficients must be proportional. If the entire equation 3x + 5y = 27 is multiplied by 4, the result is 4(3x + 5y) = 4(27), or 12x + 20y = 108. Thus, in order for the two equations to be equivalent, 3b = 108, or b = 36.

To solve this problem, multiply the expression (x + 3) by (x + s) to get x² + 3x + sx + 3s. You are given that x² + rx + 12 is equivalent to x² + 3x + sx + 3s.
Therefore, 3s is equal to 12, making s equal to 4. It is also apparent that 3x + sx is equivalent to rx.
Set the quantities equal and solve for r, as follows:
rx = 3x + sx
rx = x(3 + s)
r = 3 + s
Because s = 4,r must equal 7.

Regardless of the dimensions of a rectangle, tripling the length and width will always have the effect of tripling the perimeter because perimeter is directly proportional to length and width (P = 2l + 2w).