two sides of an equilateral trianlge measure (2y+3) units and (y^2-5) units. if the perimeter of the triangle is 33 units, what is the value of y?
To find the value of y, we can start by using the formula for the perimeter of an equilateral triangle. The perimeter of an equilateral triangle is the sum of the lengths of all three sides.
In this case, we have two sides that measure (2y+3) units and (y^2-5) units, and we know that the perimeter is 33 units.
So, we can set up the equation:
(2y+3) + (y^2-5) + (2y+3) = 33
Now let's simplify the equation:
2y + 3 + y^2 - 5 + 2y + 3 = 33
Combine like terms:
y^2 + 4y + 1 = 33
Now, let's rearrange the equation:
y^2 + 4y + 1 - 33 = 0
y^2 + 4y - 32 = 0
To solve this quadratic equation, we can factorize or use the quadratic formula. In this case, let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
where a = 1, b = 4, and c = -32
Substituting the values:
y = (-4 ± √(4^2 - 4 * 1 * -32)) / (2 * 1)
y = (-4 ± √(16 + 128)) / 2
y = (-4 ± √144) / 2
y = (-4 ± 12) / 2
Now, we have two possible solutions:
y = (-4 + 12) / 2 = 8 / 2 = 4
y = (-4 - 12) / 2 = -16 / 2 = -8
However, since a negative length is not meaningful in this context, we discard the negative solution.
Therefore, the value of y is 4.