Solving By Factoring:
The length of the side of a large square is 1 cm less than twice the length of the of the side of a smaller square. The area of the large square is 33cm^2 more than the area of the small square. Find the length of the sides of the two squares.
Please show work!!!
(2x-1)^2 = x^2 + 33
4x^2-4x+1 = x^2+33
3x^2-4x-32 = 0
(3x+8)(x-4) = 0
...
To solve this problem, we can start by assigning variables to the lengths of the sides of the squares.
Let's say the length of the side of the smaller square is 'x' cm. Therefore, the length of the side of the larger square is (2x - 1) cm, as it is 1 cm less than twice the length of the side of the smaller square.
The area of a square is calculated by squaring the length of its side. So, the area of the smaller square is x^2 cm^2, and the area of the larger square is (2x - 1)^2 cm^2.
According to the problem, the area of the larger square is 33 cm^2 more than the area of the smaller square. Hence, we can set up the following equation:
(2x - 1)^2 = x^2 + 33
To solve this equation, we can use factoring. Let’s expand the left side of the equation:
4x^2 - 4x + 1 = x^2 + 33
Now, we can re-arrange the terms to isolate the variable on one side:
4x^2 - x^2 - 4x - 33 + 1 = 0
Combining like terms, we get:
3x^2 - 4x - 32 = 0
To factor this quadratic equation, we look for two binomials whose product equals 3x^2 - 4x - 32. The factors will have the form (x + a)(x + b).
We need to find two numbers whose product is -96 (the constant term) and whose sum is -4 (the coefficient of 'x'). After considering the options, we see that the numbers -12 and 8 satisfy this condition.
Thus, we can factor the quadratic equation:
(x - 12)(3x + 8) = 0
Now, we can solve for 'x' by setting each factor equal to zero:
x - 12 = 0 or 3x + 8 = 0
Solving these two equations, we find:
x = 12 or x = -8/3
Since the length of a side cannot be negative, we discard the second solution, x = -8/3.
Hence, the length of the side of the smaller square is x = 12 cm. Therefore, the length of the side of the larger square is 2x - 1 = 2(12) - 1 = 23 cm.
Therefore, the length of the sides of the two squares is 12 cm and 23 cm, respectively.