Find all values of p such that 2(x+4)(x-2p) has a minimum value of -18.
(AoPS, Please show steps)
Why guys?
Okay, i just saw another post u made "Aops". Yes i know who you guys are. don't think i don't. ur math campus or whatever. look at this link: www.jiskha.com/questions/1577346/in-trapezoid-abcd-overline-ab-is-parallel-to-overline-cd-ab-7-units-and
Really AOPS, really? again? and GET THIS: here's a link of AOPS, the same user here, cheating! the very thing they're making a fuss about!
Link:www.jiskha.com/questions/1798144/point-g-is-the-midpoint-of-the-median-xm-of-xyz-point-h-is-the-midpoint-of-xy-and-point
Really this is just disappointing.
Also now that i think about it the real aops wouldn't d something like that, so my conclusion is u are not aops!
To find the values of p such that the expression 2(x+4)(x-2p) has a minimum value of -18, we need to use some properties of quadratic functions.
Step 1: Expand the expression
Multiply the terms using the distributive property to expand the expression:
2(x + 4)(x - 2p) = 2x(x) + 2x(-2p) + 4(x) + 4(-2p)
Simplify:
2x^2 - 4px + 4x - 8p
Step 2: Combine like terms
Combine the x terms and the constant terms:
2x^2 + (4 - 4p)x - 8p
Step 3: Identify the quadratic function
The expression obtained, 2x^2 + (4 - 4p)x - 8p, is a quadratic function in standard form, written as ax^2 + bx + c, where a = 2, b = (4 - 4p), and c = -8p.
Step 4: Find the minimum value
To find the minimum value of the quadratic function, we can use the formula for the x-coordinate of the vertex, given by x = -b/2a.
In this case, the x-coordinate of the vertex corresponds to the value of x that will make the quadratic function have its minimum value.
So, x = -(4 - 4p)/(2 * 2) = -(4 - 4p)/4 = -(1 - p).
Step 5: Substitute the x-coordinate into the quadratic function
Substitute the value of x into the quadratic function to find the minimum value:
f(-(1 - p)) = 2(-(1 - p))^2 + (4 - 4p)(-(1 - p)) - 8p
Step 6: Simplify the expression
Simplify the quadratic function to obtain the minimum value:
f(-(1 - p)) = 2(1 - 2p + p^2) + (-4 + 4p)(-1 + p) - 8p
= 2 - 4p + 2p^2 - 4 + 8p + 4p - 4p^2 - 8p
= 2 - 4 + 8p + 4p + 8p - 4p^2 - 8p + 2p^2 - 4
= -4p^2 + 14p - 6p^2 - 10
Step 7: Set the minimum value equal to -18
We want the minimum value of the quadratic function to be equal to -18. So, we solve the equation:
-4p^2 + 14p - 6p^2 - 10 = -18
-10p^2 + 14p - 10 = -18
-10p^2 + 14p + 8 = 0
Step 8: Solve the quadratic equation
To solve the quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, factoring would be a bit complicated, so we'll solve it using the quadratic formula.
The quadratic formula states that for a quadratic equation ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -10, b = 14, and c = 8. Substituting these values, we can solve for p.
p = (-14 ± √(14^2 - 4*(-10)*8)) / (2*(-10))
p = (-14 ± √(196 + 320)) / (-20)
p = (-14 ± √516) / (-20)
Step 9: Simplify the solutions
To simplify, we can further simplify the square root and divide both the numerator and denominator by their greatest common divisor, which is 4.
p = (-14 ± √(4 * 129)) / (-20)
p = (-14 ± 2√129) / (-20)
p = (-7 ± √129) / (-10)
Hence, the values of p such that the expression 2(x+4)(x-2p) has a minimum value of -18 are p = (-7 + √129) / (-10) and p = (-7 - √129) / (-10).
2(x+4)(x-2p) = 2(x^2 + (4-2p)x - 8p)
Completing the square, we have
2(x + (2-p))^2 - 8p - (2-p)^2)
So, at x = p-2, the value is
2(-8p-(2-p)^2) = -2(p+2)^2
If that is -18,
(p+2)^2 = 9
p+2 = ±3
p = -5 or 1
check:
2(x+4)(x+10) = 2(x+7)^2 - 18
2(x+4)(x-2) = 2(x+1)^2 - 18
Both have vertex at y = -18