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# I am having a problem solving 2x^2-9x+4

I would be able to do problem if the 4 had a x but i cant breakdown otherwise

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)

This problem is factorable since 2*4 = (-1)(-8), and (-1) + (-8) = -9.

2x^2-9x+4 = (2x-1)(x-4) = 0

The expression you have given is not an equation (does not equal something), so it cannot be solved.  I am going to assume that the expression is supposed to equal zero, and then solve that equation.

The equationthat I am going to solve is 2x2 - 9x + 4 = 0.  The method I am going to use is called completing the square.

2x2 - 9x + 4 = 0

2x2 - 9x + 4 -4 = 0 -4              Subtract 4 from each side

2x2 - 9x = -4                               Simplify

(2x2 - 9x)/x = -4/2                      Divide both sides by two (the coefficient of x2)

x2 -9/2*x = -2                            Distribute on the left, simplify the right

x2 - 9/2*x + 81/16 = -2 + 81/16   Add 81/16 to each side (divide the coefficient on x by two, then square

the result)

(x - 9/4)2 = -2 + 81/16                  Factor the left side as a perfect square (9/4 is the square root of

81/16)

(x - 9/4)2 = -32/16 + 81/16           Change the fractions on the right so that they have common

denominators

(x - 9/4)2 = 49/16                        Simplify

x - 9/4 = +/- sqrt(49/16)              Take the square root of both sides (sqrt means square root)

x - 9/4 = +/- 7/4                         Simplify

x - 9/4 = 7/4 or x -9/4 = -7/4       Separate the problem into positive and negative values on the right

x - 9/4 + 9/4 = 7/4 +9/4 or          Add 9/4 to both sides on both possible solutions

x - 9/4 + 9/4 = -7/4 + 9/4

x = 16/4 or x = 2/4                     Simplify each solution

x = 4 or x = 1/2                          Reduce the fractions

Completing the square is a valuable tool to use, because any binomial equation can be solved with it.  Even though you could find the answer to this problem without it, I wanted to give you a chance to learn how to use it.

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