Quadratic Equations

Quadratic Equations: Level 3 Challenges


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 aa and bb are non-zero real numbers, simplify: (a+1a)2+(b+1b)2+(ab+1ab)2(a+1a)(b+1b)(ab+1ab)\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 x(x+1)(x+2)(x+3)=120

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

(x25x+5)(x211x+30)=1\large (x^2 - 5x + 5)^{(x^2 - 11x + 30)} = 1

How many different integers satisfy the equation above?

How many real value(s) of xx exist satisfy the equation (5+26)x23+(526)x23=10? \left( 5+2\sqrt{6} \right)^{x^2-3}+\left( 5-2\sqrt{6} \right)^{x^2-3}=10 ?


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