To find a complete factorisation of 2x^2 + 7x + 3, you need to find all solutions of the equation
2x^2 + 7x + 3 = 0.
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This can be done e.g. with "completion of the circle".
Let me show you how it's done!
... compute -3 on both sides of the equation...
<=> 2x^2 + 7x = -3
... divide by 2 on both sides of the equation ...
<=> x^2 + 7/2 x = - 3/2
Now, the goal is to create a quadratic expression like x^2 + 2ax + a^2 on the left side of the equation.
We already have x^2 and 7/2 x = 2ax which means that a = 7/4. So, to complete the circle, we need to add (7/4)^2 to the expression on the left side.
However, as we don't want to destroy the equality, it is also needed to add the same expression on the right side, too.
<=> x^2 + 7/2 x + (7/4)^2 = - 3/2 + (7/4)^2
As next, due to the rule x^2 + 2ax + a^2 = (x+a)^2, you can transform the left side.
<=> (x + 7/4)^2 = - 3/2 + (7/4)^2
<=> (x + 7/4)^2 = 25/16
Now, you can draw the root on both sides.
Be careful though: you will have two solutions since both (5/4)^2 = 25/16 and (-5/4)^2 = 25/16 hold.
x + 7/4 = 5/4 color(white)(xx)" or " color(white)(xx) x + 7/4 = - 5 / 4
x = - 1/2 color(white)(xxxiii)" or " color(white)(xx) x = -3
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With your solutions x = -1/2 and x = -3 you have:
color(white)(xxx) 2x^2 + 7x + 3 = 0
<=> (x- (-1/2))(x-(-3)) = 0
<=> (x + 1/2)(x+3) = 0
Thus,
2x^2 + 7x + 3 = (x + 1/2)(x+3).
