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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.

Determining Types of Roots

Which of the following equations has two distinct real roots?

\[\begin{array} &(A)~x^2-x+1=0 &&(B)~16x^2+8x+1=0\\ (C)~\frac{2}{5}x^2+\frac{3}{10}x+\frac{1}{5}=0 &&(D)~x^2+5x+4=0 \end{array}\]

If \(a < 3,\) which of the following is true about the quadratic equation: \[x^2+2ax+a^2-3a+11=0 ?\]

If the quadratic equation \[x^2+7x-k=0\] has real roots, and the quadratic equation \[x^2-2x-k=0\] has non-real complex roots, what is the range of \(k?\)

How many distinct real roots does the following equation have: \[(x^2+x+4)(x^2+6x+8)=0?\]

Which of the following is true about the quadratic equation: \[41x^2-20x+4=5x^2+4x?\]

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