Question

Solve the quadratic equation x2 - 4x + 57 = -5 by completing the square.
A. x equals one plus or minus i times the square root of fifty
B. x equals two plus or minus the square root of fifty eight
C. x equals two plus or minus i times the square root of fifty eight
D. x equals two plus or minus the square root of sixty two

Answer 1

\( \large x^2-4x+57 = -5\\
\large x^2-4x+62 = 0\)
complete the square by adding and subtracting \(\displaystyle \left(\frac{b}{2a}\right)^2=\left( \frac{-4}{2(1)}\right)^2=4\)
\[\large \underbrace{x^2-4x+4}-4+62=0\\
\large \underbrace{(x-2)^2}+58=0 \]
Consider this as a difference of squares (i.e. \(a^2-b^2 =(a-b)(a+b)\) ) by re-writing it as:
\[\large (x-2)^2-(-58)=0\\
\large [(x-2)-\sqrt{-58}][(x-2)+\sqrt{-58}]=0\\
\large (x-2-i\sqrt{58})(x-2+i\sqrt{58})=0\]
Hence:
\(\large x=2+i\sqrt{58}\)
\(\large x = 2 - i\sqrt{58}\)