All Questions ⁺⁰237734 Questions

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Algebra

If (x,y) satisfies the simultaneous equations
3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.
x^2 - y^2 = 7 + 4xy
where x and y may be complex numbers, determine all possible values of y^2.

Jasmo Feb 26, 2025

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20

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+456

Algebra

Factor 3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.

Jasmo Feb 26, 2025

Feb 25, 2025

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29

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+456

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

Jasmo Feb 25, 2025

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33

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+456

Algebra

Fill in the blanks, to complete the factorizaion:
(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)

Jasmo Feb 25, 2025

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39

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+456

Algebra

Factor x^2 - 2x + 7 - y^2 - 10yz - 13z^2 as the product of two polynomials of degree 1.

Jasmo Feb 25, 2025

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28

2

+456

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 7n - 3mn + 15m + 8n + ___

Jasmo Feb 25, 2025

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+357

Algebra

Factor x^2 - 2x - y^2 + 2yz + 5z^2 as the product of two polynomials of degree 1.

cwimie Feb 25, 2025

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36

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+357

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 5n - mn + 20m + 4n + ___

cwimie Feb 25, 2025

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24

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+357

Algebra

Let r, s, and t be solutions of the equation 3x^3 - 4x^2 - 2x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}.

cwimie Feb 25, 2025

+1

37

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+357

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

cwimie Feb 25, 2025

+1

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+357

Geometry

Let $G$ be the center of equilateral triangle $XYZ.$ A dilation centered at $G$ with scale factor $-1$ is applied to triangle $XYZ,$ to obtain triangle $X'Y'Z'.$ Let $A$ be the area of the region that is contained in both triangles $XYZ$ and $X'Y'Z'.$ Find read more ..

cwimie Feb 25, 2025

+1

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+357

Geometry

Points $A$ and $B$ are on side $\overline{YZ}$ of rectangle $WXYZ$ such that $\overline{WA}$ and $\overline{WB}$ trisect $\angle ZWX$. If $WX = 2$ and $XY = 3$, then what is the area of rectangle $WXYZ$?

cwimie Feb 25, 2025

+1

42

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+357

Geometry

In rectangle $WXYZ$, $A$ is on side $\overline{WX}$, $B$ is on side $\overline{YZ}$, and $C$ is on side $\overline{XY}$. If $AX = 15$, $BY = 20$, $\angle ACB= 60^\circ$, and $CY = 3 \cdot CX$, then find $AB$.

cwimie Feb 25, 2025

+2

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+38

Trig help (no random guests allowed)

Problem:
Find AC.
[asy] size(300); pair A,B,C,O; A=origin; B=6*sqrt(3)*dir(150); C=(5,0);draw(A--B--C--cycle); dot("$A$",A,SW); dot("$B$",B,NW); dot("$C$",C,SE); label("$150^\circ$",A,dir(75)); label("$9$",A--B,SW); label("$7\sqrt{3}$",C--B,NE);read more ..

skibidibrain901 Feb 25, 2025

Feb 24, 2025

+2

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+17

HELP ME!!111

The graph shows that Anna and Boris begin walking at the same time. Find Boris's speed. Edit: I know that this forum is mostly about the Aops class ciriculum.This problem comes from a different ciriculum. As a new member who lacks the knowledge of this read more ..

Chiccen Feb 24, 2025

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16

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+456

Algebra

Suppose r and s are the values of x that satisfy the equation
x^2 - 2mx + (m^2 - 6m + 11) = 0
for some real number m. Find the minimum real value of (r - s)^2.

Jasmo Feb 24, 2025

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20

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+456

Algebra

Let r, s, and t be solutions of the equation x^3 + 2x^2 - 5x + 15 = 0. Compute
\frac{1}{r - 2s - 2t} + \frac{1}{s - 2r - 2t} + \frac{1}{t - 2r - 2s}

Jasmo Feb 24, 2025

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21

3

+456

Algebra

Let r, s, and t be solutions of the equation x^3 - 4x^2 - 7x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}

Jasmo Feb 24, 2025

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34

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+456

Algebra

Let t be a root of f(x) = x^3 - x + 3. Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.

Jasmo Feb 24, 2025

+1

19

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+357

Geometry

Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point.
An equilateral triangle, a regular decagon, and a regular n-gon, all with the same side length, also completely surround a read more ..

cwimie Feb 24, 2025

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+357

Geometry

Quadrilateral ABCD is a parallelogram. Let E be a point on \overline{AB}, and let F be the intersection of lines DE and BC. The area of triangle EBF is 1, and the area of triangle BCE is 2. Find the area of parallelogram ABCD.

cwimie Feb 24, 2025

+1

33

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+357

Geometry

In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH.

cwimie Feb 24, 2025

0

37

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+357

Geometry

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$ read more ..

cwimie Feb 24, 2025

+4

1

19

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+122

algebra!

what is the answer 38x+74-92=-18

thatgirlaj Feb 24, 2025

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