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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.

Repeated Roots

         

If the quadratic equation

\[ ax^2 + 2bx + c = 0 \]

has 2 repeated roots, which of the following equations must be true?

If the graph of the quadratic function \( f(x)= x^2 + 14x + a\) is tangent to the \(x\)-axis, what is \(a?\)

If the quadratic equation \[f(x) = x^2-2(k-a)x+k^2+a^2-4k-b+15\] has a repeated root for all values of \(k\), what is the value of \(a+b\ \)?

How many integers \(a\) are there such that at least one of the two quadratic equations \[x^2-2ax+9=0, x^2-2ax+11a+12=0\] has complex, non-real roots?

If the quadratic \(kx^2-16x+1\) is a perfect square polynomial, what is the value of \(k\)?

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