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Quadratic Discriminant

This one number can tell you whether the solutions to a quadratic equation are real or non-real, and whether they are distinct or repeated.

Range of Solutions

How many intersection points are there between the two graphs of the functions \[y=2x^2-4x, y=3x+5?\]

If real numbers \(x\) and \(y\) satisfy the equation \[x^2+y^2-2x-14y+50=0,\] what is the value of \(x+y\)?

Consider two sets \[\begin{align} & A=\{(x,y) \mid y=x^2-2x+35\}, \\ & B=\{(x,y) \mid y=2kx-1\}. \end{align} \] How many integers \(k\) satisfy \(A \cap B=\emptyset\)?

Find the values of \(b\) such that \(y=bx\) is a secant line to the curve \(y=2x^2-b\).

Consider the graphs of \[y=x^2+ax-5, y=2x-6.\] If they intersect at only one point, what is the value of the positive number \(a\)?

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