Akhain1

258 Questions
16 Answers

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Quadratic Functions

What real value of $t$ produces the smallest value of the quadratic $t^2 -9t - 36 + 13t - 60$?

Akhain1 May 14, 2024

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Quadratic Function

If s is a real number, then what is the smallest possible value of 2s^2 - 8s + 19 - 7s^2 - 8s + 20?

Akhain1 May 14, 2024

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Roots of Quadratic

Let a and b be the solutions to 7x^2 + x - 5 = 6x^2 - 5x + 12. Compute 1/(a^2 - 4) + 1/(b^2 - 4).

Akhain1 May 14, 2024

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Quadratic Functions

If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4 +5t^2 - 4t + 18$?

Akhain1 May 14, 2024

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Algebra

Let x and y be complex numbers. If x + y = 2 and xy = 5 - x^2 - y^2, then what is x^2 + y^2?

Akhain1 May 14, 2024

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Algebra

Let a and b be complex numbers. If a + b = 4 and a^2 + b^2 = 6 + ab, then what is a^3 + b^3?

Akhain1 May 14, 2024

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Algebra

Let f(x) = \sqrt{x}
Find the largest three-digit value of x such that f(x) is an integer.

Akhain1 May 14, 2024

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Algebra

Compute
\frac{1}{1 \times 4} + \frac{1}{7 \times 10}

Akhain1 May 14, 2024

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Palindromes

A palindrome is a number that is the same when read forwards and backwards, such as 43234. Find the number of 4-digit palindromes that are divisible by 8.

Akhain1 May 13, 2024

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Coordinates

In the diagram, the four points have coordinates $A(0,1)$, $B(1,3)$, $C(5,2)$, and $D(4.5,0)$. What is the area of quadrilateral $ABCD$?

Akhain1 May 13, 2024

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Coordinate geometry

Determine the coordinates of the point $P$ on the line $y=-x+6$ such that $P$ is equidistant from the points $A(10,-12)$ and $O(2,8)$ (that is, so that $PA=PO$). Express your answer as an ordered pair $(a,b)$.

Akhain1 May 13, 2024

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Coordinates

In triangle $ABC$, point $X$ is on side $\overline{BC}$ such that $AX = 13,$ $BX = 13,$ $CX = 5,$ and the circumcircles of triangles $ABX$ and $ACX$ have the same radius. Find the area of triangle $ABC$.

Akhain1 May 10, 2024

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Coordinates

Triangle $ABC$ has vertices $A(2, 3),$ $B(1, -8)$ and $C(-3, 2).$ The line containing the altitude through $A$ intersects the point $(0, y).$ What is the value of $y$?

Akhain1 May 10, 2024

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Coordinates

The equation of a circle in the coordinate plane can be written as $Ax^2 + 2y^2 + Bx + Cy = 50.$ The center of the circle is at $(-5,2)$. Let $r$ be the radius of the circle. Find A+B+C+r.

Akhain1 May 10, 2024

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Geometry

Find the area of the region enclosed by the graph of the equation x^2-14x+3y+70=15+9y-y^2 that lies below the line y=x-3.

Akhain1 May 10, 2024

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Geometry

Let k be a positive real number. The line x + y = 3k and the circle x^2 + y^2 = k - 4x - 6y are drawn. Find $k$ so that the line is tangent to the circle.

Akhain1 May 10, 2024

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Algebra

For certain values of $k$ and $m,$ the system
a + 2b = -3 - 7a + b,
4a + 2b = k - 5a - mb
has infinitely many solutions $(a,b).$ What are $k$ and $m?$

Akhain1 May 10, 2024

0

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Algebra

Find the ordered pair (s, t) that satisfies the system
(s/2) + 5t =3 - 7t + 8s
3t - 6s = 9 - 2t

Akhain1 May 10, 2024