Use the given area A of the rectangle to find the value of x, then give the dimensions of the rectangle.
A=209
(4x+3)(4x-5)=209
What do I do now?
16x^2 - 8x - 15 = 209
now solve in the usual way.
Amazing help, thankyou
To find the value of x and the dimensions of the rectangle, you need to solve the equation (4x+3)(4x-5) = 209. Here's how you can do that:
Step 1: Expand the equation:
16x^2 - 20x + 12x - 15 = 209
Step 2: Simplify the equation:
16x^2 - 8x - 15 = 209
Step 3: Move all the terms to one side of the equation:
16x^2 - 8x - 15 - 209 = 0
Step 4: Combine like terms:
16x^2 - 8x - 224 = 0
Step 5: Factor the equation (if possible) or use the quadratic formula to solve for x.
Step 6: Solve the equation:
The factored form of the equation is (4x + 14)(4x - 16) = 0.
Setting each factor equal to zero and solving for x:
4x + 14 = 0 or 4x - 16 = 0
Solving these equations, you get:
4x = -14 or 4x = 16
x = -14/4 or x = 16/4
x = -7/2 or x = 4
Step 7: Determine the dimensions of the rectangle:
Since x cannot be negative in this case (as it represents the length of a rectangle), the value of x is 4.
The dimensions of the rectangle are:
Length = 4x + 3 = 4(4) + 3 = 19
Width = 4x - 5 = 4(4) - 5 = 11
To find the value of x and the dimensions of the rectangle, you need to solve the given quadratic equation (4x+3)(4x-5) = 209. Here are the steps to solve it:
1. Expand the equation: (4x+3)(4x-5) = 209
=> 16x^2 - 20x + 12x - 15 = 209
=> 16x^2 - 8x - 224 = 0
2. Rearrange the equation to standard quadratic form (ax^2 + bx + c = 0), dividing all terms by the leading coefficient (16 in this case):
=> x^2 - 0.5x - 14 = 0
3. To solve the quadratic equation, you can factor it or use the quadratic formula. In this case, the equation does not easily factor, so you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation: a = 1, b = -0.5, and c = -14
4. Substitute these values into the quadratic formula and simplify:
x = (-(-0.5) ± √((-0.5)^2 - 4(1)(-14))) / (2(1))
=> x = (0.5 ± √(0.25 + 56)) / 2
=> x = (0.5 ± √(56.25)) / 2
=> x = (0.5 ± 7.5) / 2
5. Now, calculate the two possible solutions for x:
a) When x = (0.5 + 7.5) / 2 = 8 / 2 = 4
b) When x = (0.5 - 7.5) /2 = -7 / 2 = -3.5
6. Now that you have the values of x, substitute them back into the expression (4x+3) to find the dimensions of the rectangle:
a) When x = 4:
Dimension 1: 4x + 3 = 4(4) + 3 = 19
Dimension 2: 4x - 5 = 4(4) - 5 = 11
So, the dimensions of the rectangle are 19 units by 11 units.
b) When x = -3.5:
Dimension 1: 4x + 3 = 4(-3.5) + 3 = -14 + 3 = -11
Dimension 2: 4x - 5 = 4(-3.5) - 5 = -14 - 5 = -19
Since dimensions cannot be negative, these values are not valid for a rectangle.
Therefore, the value of x is 4, and the dimensions of the rectangle are 19 units by 11 units.