Question

Express


`2x {}^(2) - 20x + 53`
In the form

`2(x - p) {}^(2) + q`
Where p and q are integers.

The answer to this is

`2(x - 5) {}^(2) + 3`
Use this result to explain why the equation

`2x {}^(2) - 20x + 53 = 0`
has no real roots

Answer 1

`ax^2+bx+c=a(x-p)^2+q\ where\\\\p=(-b)/(2a);\ q=f(p)=(-(b^2-4ac))/(4a)`
We have:

`2x^2-20x+53\to a=2;\ b=-20;\ c=53`
substitute

`p=(-(-20))/(2\cdot2)=(20)/(4)=5\\\\q=f(5)=2\cdot5^2-20\cdot5+53=2\cdot25-100+53=50-100+53=3`
therefore we have the answer:

`2x^2-20x+53=2(x-5)^2+3`