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33 problems match “quadratic equations”. Clear

Nested rational equation with a single root ●●○○○ 20 ptsSolve the equation $x+\frac{x}{x}+\frac{x}{x+\frac{x}{x}}=1$ and give the value of $x$ .
Ratio of sums in a geometric sequence ●●○○○ 20 ptsA geometric sequence $\{a_n\}$ has all positive terms and satisfies $a_3 = a_1 + 2a_2$ . Find the value of $\frac{a_9+a_{10}}{a_7+a_8}$ . Round to two decimal places.
Equation with reciprocals of linear factors ●●○○○ 20 ptsFind the roots of $\frac{1}{x+4} - \frac{1}{x-7} = \frac{11}{30}$ , with $x \neq -4$ and $x \neq 7$ .
Difference of fractions equals one ●●○○○ 20 ptsFind all values of $x$ (with $x \neq 0, 1, -1$ ) for which $\frac{1}{x^2 - x} - \frac{1}{x^3 - x} = 1$ . Round each value to two decimal places.
Width of a parabolic arch base ●●○○○ 20 ptsA parabolic arch is 18 m high, and its axis is vertical. The width of the arch 8 m below the top is 64 m. Find the width of the arch at its base.
Points where f equals its average value ●●○○○ 20 ptsLet $f(x) = (x-2)^2$ . Find all values of $c$ such that $f(c)$ equals the average value of $f$ over the closed interval $[-1, 5]$ . Round each answer to two decimal places.
Garden dimensions from perimeter and area ●●○○○ 20 ptsA rectangular garden has a perimeter of 68 ft and an area of 240 ft $^2$ . Find its length and width. Give the width (the smaller value) first, then the length.
Length and width of an accent wall ●●○○○ 20 ptsA wall has a length that is 3 ft more than its width. The wall has an area of 130 ft². Find the length and the width of the wall, in feet. Round to three decimal places.
Length of a rectangular patio ●●○○○ 20 ptsA rectangular patio has an area of 432 square feet. Its width is $\frac{3}{4}$ of its length. Find the length of the patio in feet.
Largest component reliability under budget ●●○○○ 20 ptsA component's cost is $C = (10P)^2$ dollars, where $P$ is the probability that the component operates as expected. A system has two identical components, and both must operate for the system to operate. A total of $173 can be spent on the two components. What is the largest component probability $P$ that can be achieved? Round to two decimal places.