Quadratics

Solve for x:
2 x^2-8 x+7=0

 

Divide both sides by 2:
x^2-4 x+7/2=0

 

Subtract 7/2 from both sides:
x^2-4 x=-7/2

 

Add 4 to both sides:
x^2-4 x+4=1/2

 

Write the left hand side as a square:
(x-2)^2=1/2

 

Take the square root of both sides:
x-2=1/sqrt(2) or x-2=-1/sqrt(2)

 

Add 2 to both sides:
x=2+1/sqrt(2) or x-2=-1/sqrt(2)

 

Add 2 to both sides:
Answer: |  x=2+1/sqrt(2)      or          x=2-1/sqrt(2)

 

Youn can finish the 2nd part.

Let \(\alpha \) and \(\beta\) be the 2 roots of the equation ax²+bx+c

\(\alpha + \beta = -\frac{b}{a}\)

\(\alpha \beta = \frac{c}{a}\)

\(\frac{1}{2\alpha}+\frac{1}{2\beta}\)

=\(\frac{\alpha+\beta}{2\alpha\beta}\)

=\(\frac{-\frac{-8}{2}}{2(\frac{7}{2})}\)

=\(\frac{4}{7}\)