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33 problems match “quadratic equations”. Clear

Solving a quadratic with exponents ●●○○○ 20 ptsSolve $4(3 - x) + 9^2 = x(x - 4) + 7^2 - 5$ for the positive value of $x$ .
Third term of a power-plus-constant sequence ●●○○○ 20 ptsLet $\{a_n\}$ be the sequence given by $a_n = a^n + m$ for positive integers $n$ , where $a < 0$ and $m$ are constants. Given $a_1 = 2$ and $a_2 = 4$ , find $a_3$ .
Smallest modulus in a complex system ●●●○○ 50 ptsThe complex numbers $z$ and $w$ satisfy the system $\begin{align*} z + \frac{20i}{w} &= 5+i, \\ w+\frac{12i}{z} &= -4+10i. \end{align*}$ Find the smallest possible value of $|zw|^2$ .
Two numbers from mean conditions ●●●○○ 50 ptsFind two numbers whose arithmetic mean exceeds their geometric mean by $\frac{1}{2}$ and whose harmonic mean is $\frac{25}{26}$ less than their arithmetic mean.
Maximum number of quadratic equations on a board ●●●○○ 50 ptsWhat is the maximum number of different monic quadratic equations that can be written on a board, given that any two of them have a common root, but no four of them have a root common to all?
Unique geometric sequence with shifted terms ●●●○○ 50 ptsThe geometric sequence $\{a_n\}$ satisfies $a_1=a$ with $a>0$ , and $a_1+1$ , $a_2+2$ , $a_3+3$ form a geometric sequence. If $\{a_n\}$ is unique, find the value of $a$ . Round to two decimal places.
Ratio in a geometric sequence ●●●○○ 50 ptsIn the geometric sequence $\{a_n\}$ of real numbers, $a_5 \cdot a_{11} = 4$ and $a_3 + a_{13} = 5$ . Find all possible values of $\frac{a_{14}}{a_4}$ . Give each value as a decimal, in increasing order.
Two numbers from three means ●●●○○ 50 ptsFind two positive numbers whose arithmetic mean exceeds their geometric mean by $2$ and whose harmonic mean is $3.6$ less than their arithmetic mean. Give them in increasing order.
Percent of A and A+B equal ●●●○○ 50 ptsLet $A$ and $B$ be positive numbers. $B$ as a percent of $A$ is equal to $A$ as a percent of $(A+B)$ . Find $B$ as a percent of $A$ . Round to the nearest whole percent.
Coefficient from line-plane angle ●●●○○ 50 ptsThe angle between the plane $3x + by - 5z = 2$ and the straight line $x + 2 = \frac{3 - y}{2} = \frac{z - 1}{2}$ is $45^\circ$ . Find the constant $b$ , given that $0 < b < 10$ .