Use the given area A of the rectangle to find the value of x. Then give the dimensions of the rectangle.
A=91m^2
A=L•W
L=(2x+3)m
W=(x+2)m
91m^2=(2x+3)m•(x+2)m
91=(x+2)(2x+3)
Now what?
Now it's time to solve the equation and find the value of x. But before we continue, let me tell you a picture joke to lighten the mood:
Why don't scientists trust atoms?
Because they make up everything!
Now back to the problem. Let's solve the equation:
91 = (x+2)(2x+3)
Expanding the brackets:
91 = 2x^2 + 7x + 6
Rearranging into a quadratic equation:
2x^2 + 7x + 6 - 91 = 0
2x^2 + 7x - 85 = 0
Now we can either solve this quadratic equation by factoring, completing the square, or using the quadratic formula. But before we proceed, here's another quick joke for you:
Why don't skeletons like parties?
Because they have no body to dance with!
Alright, now let's solve the quadratic equation to find the value of x.
To find the value of x, you need to solve the equation:
91 = (x + 2)(2x + 3)
To solve the quadratic equation, you can start by expanding the equation:
91 = 2x^2 + 7x + 6x + 6
Combine like terms:
91 = 2x^2 + 13x + 6
Rearrange the equation to have zero on one side:
2x^2 + 13x + 6 - 91 = 0
Simplify further:
2x^2 + 13x - 85 = 0
Now, you can solve this quadratic equation. You can factor it or use the quadratic formula to find the values of x.
Factoring the equation, you need to find two numbers that multiply to give AC (2 * -85 = -170) and add up to B (13). The numbers are 17 and -10.
Now rewrite the equation using these numbers:
2x^2 + 17x - 10x - 85 = 0
Factor by grouping:
x(2x + 17) - 5(2x + 17) = 0
Using the zero product property, set each factor equal to zero:
x = 0 or 2x + 17 = 5
If x = 0, then it doesn't make sense in this context, as it would result in one of the dimensions being zero.
Solving for 2x + 17 = 5:
2x + 17 - 17 = 5 - 17
2x = -12
x = -12/2
x = -6
Since the dimensions of a rectangle cannot be negative, x = -6 is not a valid solution.
Therefore, in this case, there is no real value of x that satisfies the given equation.
To find the value of x and the dimensions of the rectangle, you need to solve the equation 91 = (x+2)(2x+3).
You can do this by expanding the equation and setting it equal to zero:
91 = 2x^2 + 7x + 6
0 = 2x^2 + 7x + 6 - 91
0 = 2x^2 + 7x - 85
Now you have a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula.
One way to solve it is by factoring the quadratic equation:
0 = (2x - 5)(x + 17)
This gives you two possible solutions:
1) 2x - 5 = 0, which means 2x = 5 and x = 5/2 or 2.5
2) x + 17 = 0, which means x = -17
Since the dimensions of a rectangle cannot be negative, you can discard the solution x = -17. Therefore, the value of x is 2.5.
To find the dimensions of the rectangle, substitute the value of x back into the equations for length (L) and width (W):
L = (2x + 3)m = (2(2.5) + 3)m = 8m
W = (x + 2)m = (2.5 + 2)m = 4.5m
So, the dimensions of the rectangle are 8m for length and 4.5m for width.
Set the equation equal to 0:
2x^2 + 4x + 3x + 6 - 91 = 0
2x^2 + 7x - 85 = 0
Factor:
(2x + 17) (x - 5) = 0
Set the factors equal to 0:
2x + 17 = 0; x = -17/2
x - 5 = 0; x = 5
You can't have a negative number for this problem, so use x = 5 for the value of x.
I'll let you take it from here to determine the dimensions of the rectangle.