All Questions ⁺⁰236048 Questions

0

20

1

+1310

Algebra

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

learnmgcat Jul 14, 2024

0

13

1

+1310

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.

learnmgcat Jul 14, 2024

0

14

1

+1310

Algebra

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

learnmgcat Jul 14, 2024

0

18

0

+1310

Algebra

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

learnmgcat Jul 14, 2024

0

6

4

+63

Polynomial help

This may look like a lot but I think it can be simplified down to three terms.
Factor as much as possible.

MeldHunter Jul 14, 2024

0

14

1

+893

Algebra

Let
f(x) = \frac{1}{1+\frac{2}{1+\frac 3x}}.
There are three real numbers $x$ that are not in the domain of $f(x)$. What is the sum of those three numbers?

AnswerscorrectIy Jul 14, 2024

0

20

1

+893

Algebra

The function $f(x)$ is defined only on domain $[-1,2]$, and is defined on this domain by the formula
f(x) = 2x^2-8x+1.
What is the range of $f(x)$? Express your answer as an interval or as a union of intervals.

AnswerscorrectIy Jul 14, 2024

0

14

1

+893

Algebra

For certain values of k and m, the system
3a + 2b = 2
a + 2b = k - 8a - mb
has infinitely many solutions (a,b). What are k and m? read more ..

AnswerscorrectIy Jul 14, 2024

0

14

2

+893

Algebra

Let $x$ and $y$ be nonnegative real numbers. If $x + y = 25$, then find the minimum value of $6x + 3y.$

AnswerscorrectIy Jul 14, 2024

0

15

1

+948

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.

Rangcr897 Jul 14, 2024

0

13

1

+948

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.

Rangcr897 Jul 14, 2024

0

9

1

+948

Geometry

Two circles intersect at two points, $P$ and $Q$. The equations of the two circles are $x^2 + (y - 1)^2 = 1$ and $(x - 1)^2 + y^2 = 1$. Find the length PQ.

Rangcr897 Jul 14, 2024

0

14

1

+893

Geometry

In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle CAD = 45^\circ,$ and $AD = 24,$ then find the area of triangle $ABC.$

AnswerscorrectIy Jul 14, 2024

0

8

1

+893

Geometry

In triangle $ABC,$ $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the shortest altitude in this triangle.

AnswerscorrectIy Jul 14, 2024

0

15

1

+893

Geometry

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $BC = 15,$ and $CX = 5$. What is $BX$?

AnswerscorrectIy Jul 14, 2024

0

7

1

+893

Geometry

In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.

AnswerscorrectIy Jul 14, 2024

0

11

4

+842

Algebra

A magician makes potions by combining maple syrup from a magical maple tree with ordinary water. The magician starts with a large supply of two potions: a red potion, which is 60% magical syrup by volume (and the rest is just water), and blue potion, which read more ..

RedDragonl Jul 14, 2024

0

12

1

+842

Algebra

Find $t$ if the expansion of the product of $x^3$ and $x^2 + tx$ has no $x^2$ term.

RedDragonl Jul 14, 2024

0

17

0

+842

Algebra

There exists a polynomial $f(x)$ and a constant $k$ such that
(x^2 - 2x - 5) f(x) = 2x^4 + 19x^3 + kx^2 - 15x - 1.
What is $k?$

RedDragonl Jul 14, 2024

0

8

2

+842

Algebra

What is the coefficient of $x$ in $(x^3 + x^2 + x + 1)(x^4 - 8x^3 + 17x^2 - 23x + 14)$?

RedDragonl Jul 14, 2024

0

11

1

+893

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence. If
a_n = a_{n - 1} + a_{n - 2}
for all $n \ge 3,$ and $a_{11} = 1$ and $a_{10} = 4,$ then find $a_6.$

AnswerscorrectIy Jul 14, 2024

0

13

1

+893

Algebra

Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$

AnswerscorrectIy Jul 14, 2024

0

9

1

+893

Algebra

Find the constant k such that the quadratic 2x^2 + 3x + 8x - x^2 + 4x + k has a double root.

AnswerscorrectIy Jul 14, 2024

0

9

1

+893

Algebra

Find the vertex of the graph of the equation y = -2x^2 + 8x - 15 - 3x^2 - 14x + 25.

AnswerscorrectIy Jul 14, 2024

0

11

1

+893

Algebra

The parabola y = ax^2 + bx + c is graphed below. Find a + b + c. (The grid lines are one unit apart.)
The parabola passes through (-2,8), (0,0), and (2,8).

AnswerscorrectIy Jul 14, 2024

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