Question

if x is real, prove that the quadratic expression (i)(x-2)(x+3)+7 is always positive​

Answer 1

Answer:

here is your answer mate...

Step-by-step explanation:

(x+2)2≥0  because it is a square. So 2(x+2)2+1≥1>0.

As 2(x+2)2+1=2x2+8x+9, 2x2+8x+9>0

In general, not all quadratics will be entirely positive or entirely negative but you can always convert ax2+bx+c=a(x2+b/ax+b2/4a2)+c−b2/a=a(x+b/2a)2+(c−b2/a) The term squared will always be non-negative. If a and (c - b^2/a) are both positive or are both negative the quadratic will be either always positive or negative. If they are not both then the quadratic will be be positive for some values and negative for others.

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