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Quadratic Equations

Quadratic equations describe the motion of a baseball after it connects with a bat, and the acceleration of gravity at the Earth's orbit.

Level 3

         

You went on a 360km train ride but you are late! If the train had just been 5 km/hr faster, then the train ride would have been one hour shorter.

How fast was the train going?

If \(a\) and \(b\) are non-zero real numbers, simplify: \[\left(a + \frac1a\right)^2+\left(b + \frac1b\right)^2 + \left(ab + \frac1{ab}\right)^2 \\ - \left(a + \frac1a\right)\left(b + \frac1b\right)\left(ab + \frac1{ab}\right) \]

\[ x(x+1)(x+2)(x+3)=120\]

Find the sum of all the real roots of the equation above.

\[\large (x^2 - 5x + 5)^{(x^2 - 11x + 30)} = 1\]

How many different integers satisfy the equation above?

How many real value(s) of \(x\) exist satisfy the equation \[ \left( 5+2\sqrt{6} \right)^{x^2-3}+\left( 5-2\sqrt{6} \right)^{x^2-3}=10 ? \]

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