All Questions ⁺⁰236075 Questions

0

13

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+2653

Geometry

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

LiIIiam0216 Jun 9, 2024

0

8

1

+2653

Geometry

A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The line is x = 4, and the equation of the circle is x^2 + y^2 = 25.

LiIIiam0216 Jun 9, 2024

0

23

2

+2653

Algebra

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

LiIIiam0216 Jun 9, 2024

0

10

2

+2653

Algebra

If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 131$, what is the minimal possible value of $|a - b|$?

LiIIiam0216 Jun 9, 2024

Jun 8, 2024

0

23

1

+948

Algebra

Let $m$ be a real number. If the quadratic equation $x^2+mx+4 = 2x^2 + 17x + 8$ has two distinct real roots, then what are the possible values of $m$? Express your answer in interval notation.

Rangcr897 Jun 8, 2024

0

12

1

+948

Algebra

Find the sum of the squares of the roots of $2x^2+4x-1=x^2-8x+3$.

Rangcr897 Jun 8, 2024

0

18

1

+948

Algebra

The quadratic equation $x^2-mx+24 = 10$ has roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?

Rangcr897 Jun 8, 2024

-1

11

1

+948

Algebra

The solutions to
2x^2 - 10x + 13 = -6x^2 - 18x - 15
are a+bi and a-bi, where a and b are positive. What is a\cdot b?

Rangcr897 Jun 8, 2024

0

13

1

+1310

Algebra

The solutions to
2x^2 - 10x + 13 = -6x^2 - 18x - 15
are a+bi and a-bi, where a and b are positive. What is a\cdot b?

learnmgcat Jun 8, 2024

0

16

1

+1310

Algebra

The roots of Ax^2+Bx+1 are the same as the roots of (x - 2)(x + 2). What is A+B?

learnmgcat Jun 8, 2024

0

13

1

+1310

Algebra

Find the discriminant of the quadratic 5x^2 - 2x + 8 - 2x^2 + 6x + 2 - 2x^2 + 4x - 1.

NotThatSmart ●

learnmgcat Jun 8, 2024

0

13

1

+1310

Algebra

Find the largest integer $k$ such that the equation
5x^2 - kx + 8 - 20x^2 + 45 = 0
has no real solutions.

NotThatSmart ●

learnmgcat Jun 8, 2024

0

18

1

+2653

Algebra

Find all values of $x$ such that
\frac{x}{x - 5} = \frac{4}{2x - 4} + 2x

NotThatSmart ●

LiIIiam0216 Jun 8, 2024

0

21

4

+2653

Algebra

The roots of the quadratic equation $z^2 + bz + c = 0$ are $4 + 2i$ and $-5 + 8i$. What is $b+c$?

NotThatSmart ●●●●

LiIIiam0216 Jun 8, 2024

0

15

1

+2653

Algebra

Find all ordered pairs $(x,y)$ of real numbers such that $x + y = 10$ and $x^2 + y^2 = 32 + 2xy$.
For example, to enter the solutions $(2,4)$ and $(-3,9)$, you would enter "(2,4),(-3,9)" (without the quotation marks).

LiIIiam0216 Jun 8, 2024

0

33

1

+2653

Geometry

Let $ABCD$ be a square with side length $1.$ A laser is located at vertex $A,$ which fires a laser beam at point $X$ on side $\overline{BC},$ such that $BX = \frac{1}{2}.$ The beam reflects off the sides of the square, until it ends up at another read more ..

LiIIiam0216 Jun 8, 2024

0

12

1

+355

Cannot get the answer

There is a group of five children, where two of the children are twins. In how many ways can I distribute 8 identical pieces of candy to the children, if the twins must get an equal amount of candy?

Lilliam0216 Jun 8, 2024

0

10

2

+355

help

In how many ways can you distribute 8 indistinguishable balls among 5 distinguishable boxes, if at least one of the boxes must be empty?

Lilliam0216 Jun 8, 2024

0

14

1

+1234

Geometry

Let A1 A2 A3 ... A15 be a regular polygon. A rotation centered at $A_4$ with an angle of $\alpha$ takes $A_3$ to $A_5$. Given that $\alpha < 180^\circ$, find $\alpha,$ in degrees.

Akhain1 Jun 8, 2024

0

28

1

+1234

Geometry

Let a be a real number such that 0 < a < \frac{\pi}{2}. Show that \left(\sin(a)\right)^7 + \left(\cos ( a)\right)^7< 2*sin(a)^4*cos(a)^4.

Akhain1 Jun 8, 2024

0

21

1

+1234

Geometry

Point $B$ is the image of point $A$ under a dilation with center $O$. If $OB = 30$, and the scale factor of the dilation is $5$, then what is $OA$?

Akhain1 Jun 8, 2024

0

20

1

+1234

Geometry

The perimeter of a rectangle is $40,$ and the length of one of its diagonals is $10 \sqrt{2}.$ Find the area of the rectangle.

Akhain1 Jun 8, 2024

-2

16

1

+219

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_1 = \frac{1}{5}$ and $S_2 = \frac{1}{10},$ then find $S_{15}.$

BRAlNBOLT Jun 8, 2024

-2

27

1

+219

Algebra

In a store window, there was a box of berries having a total weight of 200 kg. The berries were 98% water, by weight. After two days in the sun, the water content of the berries was only 95%, by weight. What was the total weight of the berries read more ..

BRAlNBOLT Jun 8, 2024

0

13

1

+1310

Algebra

All members of our painting team paint at the same rate. If $10$ members can paint a $1000$ square foot wall in $10$ minutes, then how long would it take for $5$ members to paint a $1000$ square foot wall, in minutes?

learnmgcat Jun 8, 2024

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