The perimeter of the quadrilateral is 56 centimeters. Find the length of each side.
A quadrilateral has sides labeled x plus 8, x plus 3, x squared minus 3 x, and 3 x minus 3.
x+8 + x+3 + x^2-3x + 3x-3 = 56
x^2 + 2x - 48 = 0
Once you find x, then you can do the sides.
To find the length of each side of the quadrilateral, we need to solve the equation that represents the perimeter.
The perimeter of a quadrilateral is the sum of its four sides. In this case, the perimeter is given as 56 centimeters.
So, we can set up the equation:
(x + 8) + (x + 3) + (x^2 - 3x) + (3x - 3) = 56
First, we'll simplify the equation by combining like terms:
x + 8 + x + 3 + x^2 - 3x + 3x - 3 = 56
Next, we'll combine like terms again:
2x + x^2 + 5 = 56
Rearranging the equation to the standard quadratic form:
x^2 + 2x + 5 = 56
Now, we'll subtract 56 from both sides of the equation:
x^2 + 2x + 5 - 56 = 0
x^2 + 2x - 51 = 0
Now we'll factor the quadratic equation:
(x + 17)(x - 3) = 0
Setting each factor equal to zero:
x + 17 = 0 or x - 3 = 0
Solving for x in each case:
x = -17 or x = 3
Since we can't have a negative length, we'll discard -17 as a solution and take the positive solution, x = 3.
Now, we can substitute the value of x back into each side length to find their lengths:
Side 1: x + 8 = 3 + 8 = 11 centimeters
Side 2: x + 3 = 3 + 3 = 6 centimeters
Side 3: x^2 - 3x = (3)^2 - 3(3) = 9 - 9 = 0 centimeters
Side 4: 3x - 3 = 3(3) - 3 = 9 - 3 = 6 centimeters
Therefore, the lengths of the sides are:
Side 1: 11 centimeters
Side 2: 6 centimeters
Side 3: 0 centimeters
Side 4: 6 centimeters
To find the lengths of the sides of the quadrilateral, we can set up an equation using the given information.
The perimeter of a quadrilateral is the sum of the lengths of all sides. In this case, the perimeter is given as 56 centimeters.
So, we can set up the equation:
(x + 8) + (x + 3) + (x^2 - 3x) + (3x - 3) = 56
Let's simplify this equation step by step:
First, combine like terms:
x + x + x^2 - 3x + 8 + 3 - 3 = 56
Next, simplify further:
x^2 - 3x + x + x + 8 + 3 - 3 = 56
x^2 - x + 8 = 56
Now, move all the terms to one side of the equation:
x^2 - x + 8 - 56 = 0
x^2 - x - 48 = 0
At this point, we have a quadratic equation. We can solve this equation by factoring or by using the quadratic formula.
Factoring:
(x - 7)(x + 6) = 0
So, we have two possible solutions for x:
x - 7 = 0 or x + 6 = 0
Solving these equations, we find:
x = 7 or x = -6
Since the length of a side cannot be negative, we can ignore the solution x = -6.
Therefore, the length of each side of the quadrilateral is:
Side 1: x + 8 = 7 + 8 = 15 centimeters
Side 2: x + 3 = 7 + 3 = 10 centimeters
Side 3: x^2 - 3x = 7^2 - 3(7) = 49 - 21 = 28 centimeters
Side 4: 3x - 3 = 3(7) - 3 = 21 - 3 = 18 centimeters