Tim L. answered 11/01/12
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There is no need to assume the expression is equal to zero.
Instead assuming the expression is equal to an unknown y will be your best analytical guide for future advanced math classes including calculus.
2x^2 - 9x + 4 is factorable to (2x-1)(x-4) as my fellow tutors have already expounded upon.
Analytically, it's an upward parabola since there is a positive x^2 included in the expression.
If the base of the parabola is above the x axis then, then y will never equal zero so bad bad bad assumption.
Calculus is your best analytical tool. When you calculate dy/dx and set the result equal to zero, you will find the x value of the base of the parabola.
dy/dx = (4x - 9) = 0, and thus the line x = 2.25 sits on the extreme base of the upward parabola.
When y = (2x-1)(x-4) and you plug in x, you'll get y = (3.5)(-1.75) = -6.125
So the base of the parabola is located at (2.25, -6.125) where the y value of the upward parabola's extreme base is below the x axis.
Only now can you conclude there are two points on y where the parabola touches the x-axis.
There are an infinite many points along the parabola, but x = 4 and x = 1/2 when y = 0 are usually of interest.
I intentionally do not say 'or', but rather I say 'and' because of definitive conclusion there are two points where the parabola touches the x-axis (also known as when y = 0)