If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4 +5t^2 - 4t + 18$?
When you combine like terms, you get 4t2 plus some other
stuff, but the square is all we're interested in for this problem.
The 4t2 indicates that this curve is a parabola that opens .
upward. Since it opens upward, there is no maximum value,
angle PCD = 75o
Grab a scrap of paper and sketch it.
The line from P to C is the base of a isoceles triangle.
Its equal sides are a side of the square and a side of the triangle.
The big angle of the isoceles triangle is the sum of the 60o of the
equilateral triangle and the 90o of the square, for a total of 150o.
The sum of the interior angles of any triangle is 180o and so that
leaves 30o for the two small angles to share, namely 15o each.
At corner C, subtract that 15o from the 90o of the square, leaving 75o.
Kaspar is buying apples and oranges from the market. The total cost of 4 apples and 3 oranges is $11, while the total cost of 2 apples and 5 oranges is $13. What is the cost of each apple and each orange?
4A + 3G = 11 (eq 1)
2A + 5G = 13 (eq 2)
Multiply (eq 2) by 2 4A + 10G = 26 (eq 3)
Subtract (eq 1) from (eq 3) 7G = 15
G = 15 / 7 = 2.14 (one orange)
Plug the value of an orange
back into (eq 1) 4A + (3)(2.14) = 11
4A = 11 – 6.42 = 4.58
A = 4.58 / 4 = 1.14 (one apple)
(eq 1) 4 • (1.14) + 3 • (2.14) = 10.98 close enough to 11
(eq 2) 2 • (1.14) + 5 • (2.14) = 12.98 close enough to 13
The answers check out, considering that the calculations required rounding.
x2 – 5x + p + 2x2 – 8x
Combine like terms, arrange in standard form ax2 + bx + c
3x2 – 13x + p
An equation is a square (another way to say this is that
it has a double root) when its discriminant equals zero
b2 – 4ac = 0
(–13)2 – (4)(3) • p = 0
169 – 12p = 0
– 12p = – 169
p = 169 / 12
Checked that the answer is a square by using Desmos to
plot the curve of y = 3x2 – 13x + 169/12 and confirming
crosses touches the x-axis at only one point.
Find the constant $k$ such that the quadratic $2x^2 + 3x + 8k + 3x^2 + 2x + k$ has a double root.
2x2 + 3x + 8k + 3x2 + 2x + k
combine like terms, and arrange in standard form ax2 + bx + c
5x2 + 5x + 9k
an equation has a double root when its discriminant equals zero
b2 – 4ac = 0
52 – (4)(5) • 9k = 0
25 – 180k = 0
– 180k = – 25
k = 25 / 180
k = 5 / 36