If the length of a rectangle is (2x-1)cm and its width is (x+3)cm.
How do i write an expression in the form ax^2+bx+c for the area of the rectangle?
Given that the area of the rectangle is 294cm^2, determine the value of x and state the dimensions of the rectangle in cms?? want the working so i can do questions like these in the future.
Area = length x width, so
Area = (2x-1)(x+3)
expand this and you got the first part
(did you get 2x^2 + 5x - 3 ?)
now set this equal to 294
2x^2 + 5x - 3 = 294
2x^2 + 5x - 297 = 0
rather than waste time trying to factor this with these numbers, I would use the quadratic formula
-594
Well, the area of a rectangle can be found by multiplying its length and width. So, in this case, the expression for the area would be (2x-1)(x+3) cm^2.
To find the value of x, we need to solve the equation (2x-1)(x+3) = 294. Let's expand the expression and set it equal to 294:
2x^2 + 6x - x - 3 = 294
2x^2 + 5x - 3 = 294
2x^2 + 5x - 297 = 0
Now, to solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our equation, we get:
x = (-5 ± √(5^2 - 4*2*(-297))) / (2*2)
x = (-5 ± √(25 + 2376)) / 4
x = (-5 ± √(2401)) / 4
x = (-5 ± 49) / 4
So, we have two possible solutions for x:
x = (-5 + 49) / 4 = 11
x = (-5 - 49) / 4 = -13.5
Since the length and width cannot be negative, the solution x = -13.5 is not valid. Therefore, the value of x is 11.
Now, let's find the dimensions of the rectangle. We can plug the value of x = 11 into the expressions for length and width:
Length = 2x - 1 = 2(11) - 1 = 22 - 1 = 21 cm
Width = x + 3 = 11 + 3 = 14 cm
So, the dimensions of the rectangle are 21 cm by 14 cm.
To write an expression for the area of a rectangle in the form ax^2 + bx + c, we need to multiply the length and width of the rectangle.
Given that the length of the rectangle is (2x - 1) cm and the width is (x + 3) cm, the area can be expressed as:
Area = Length x Width
= (2x - 1) cm * (x + 3) cm
To simplify this expression, you need to use the distributive property of multiplication. Multiply each term in the first factor by each term in the second factor:
Area = 2x * (x + 3) - 1 * (x + 3)
= 2x^2 + 6x - x - 3
= 2x^2 + 5x - 3
Therefore, the expression for the area of the rectangle in the form ax^2 + bx + c is 2x^2 + 5x - 3.
To find the value of x and state the dimensions of the rectangle, we equate the area to 294 cm^2:
2x^2 + 5x - 3 = 294
Rearrange the equation to bring everything to one side:
2x^2 + 5x - 297 = 0
Now, we can either factorize the quadratic equation or use the quadratic formula to find the value of x. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In the given equation, a = 2, b = 5, and c = -297. Plugging these values into the quadratic formula, we get:
x = (-5 ± √(5^2 - 4 * 2 * -297)) / (2 * 2)
x = (-5 ± √(25 + 2376)) / 4
x = (-5 ± √2401) / 4
x = (-5 ± 49) / 4
Solving for x, we have two possible values:
x = (-5 + 49) / 4 = 11
x = (-5 - 49) / 4 = -13.5
Since the length and width of a rectangle cannot be negative, we disregard the negative value. Therefore, x = 11.
To find the dimensions of the rectangle, substitute the value of x back into the expressions for the length and width:
Length = 2x - 1 = 2(11) - 1 = 22 - 1 = 21 cm
Width = x + 3 = 11 + 3 = 14 cm
So, the dimensions of the rectangle are 21 cm by 14 cm.