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33 problems match “quadratic equations”. Clear

Side length of a parallelogram from its area ●○○○○ 10 ptsA parallelogram has adjacent sides of lengths $s$ units and $2s$ units forming a $45^\circ$ angle. The area of the parallelogram is $8\sqrt{2}$ square units. Find the value of $s$ . Round to two decimal places.
Product of a complex root and its conjugate ●○○○○ 10 ptsLet $z$ be a complex number that satisfies $z^2 + 3 = 0$ . Find the value of $z \cdot \overline{z}$ , where $\overline{z}$ denotes the complex conjugate of $z$ .
Common ratio of a geometric sequence ●○○○○ 10 ptsIn a geometric sequence $\{a_n\}$ where all terms are positive, $a_6 = a_5 + 2a_4$ . Find the common ratio $q$ .
Side length from heptagon area ●○○○○ 10 ptsA regular heptagon has area $78.105$ square cm. Find its side length in cm, using the area formula $A \approx 3.633912444\,x^2$ , where $x$ is the side length. Round to three decimal places.
Solve a quadratic with fractional coefficient ●○○○○ 10 ptsSolve $\dfrac{2}{3}x^2 - 4 = 12$ for $x$ . Give both solutions, rounded to two decimal places.
Analytic solutions on four quadrants ●●○○○ 20 ptsHow many analytic functions $f:\Omega\to\mathbb{C}$ satisfy $f(z)^2+3i\,f(z)+4=0$ for all $z\in\Omega$ , where $\Omega$ is the complex plane with the two coordinate axes removed? Give the number of such functions.
Area of quadrilateral from circle and hyperbola ●●○○○ 20 ptsA convex quadrilateral is drawn in the coordinate plane such that each of its vertices $(x, y)$ satisfies the equations $x^{2}+y^{2}=73$ and $xy=24$ . What is the area of this quadrilateral?
Continuity of a piecewise function ●●○○○ 20 ptsLet $f(x) = \begin{cases} 5x^2+2 & \text{if } x\le a, \\ 11x & \text{if } x>a. \end{cases}$ Find the smallest possible value of $a$ for which the graph of $y=f(x)$ is continuous (the graph can be drawn without lifting your pencil from the paper). Give your answer as a decimal.
Quadratic constraint on a shifted expression ●●○○○ 20 ptsGiven $6x^{2}-4x-3=0$ , find the value of $(x-1)^{2}+x\left(x+\frac{2}{3}\right)$ .
Erasing triples on a board ●●○○○ 20 ptsOn a board the numbers $1, 2, 3, \ldots, 27$ are written. One step consists of erasing three numbers $a, b, c$ from the board and writing in their place the number $a+b+c+n$ , where $n$ is a fixed positive integer. Determine the positive integer $n$ such that, after 13 steps, the number $n^2$ is written on the board.