Question 1
For what value of \(k\) does the following system have exactly one solution?
\[\begin{aligned} y &= x^2 \\ y &= 2x + k \end{aligned}\]
Hard - Quiz 1
Challenge yourself with systems that combine linear and nonlinear equations, demanding careful algebraic moves and sharp attention to valid solutions.
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A small business sells two products. The revenue from product A is \(R_A = x^2\) dollars and from product B is \(R_B = 6x - 8\) dollars, where \(x\) is the number of units sold of each product. At what number of units does product B first generate more revenue than product A?
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What is the product of all \(x\)-coordinates of the solutions to the following system?
\[\begin{aligned} x^2 + y^2 &= 25 \\ y &= x + 1 \end{aligned}\]
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The graphs of \(y = \dfrac{6}{x}\) and \(y = x + 1\) intersect at two points. What is the sum of the \(y\)-coordinates of these intersection points?
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Answer question 4 to unlock
A researcher models the height \(y\), in meters, of a drone and a model rocket, each at time \(t\) seconds after launch:
\[\begin{aligned} y &= -t^2 + 8t \\ y &= 2t + 5 \end{aligned}\]
For how many seconds is the rocket above the drone?