Question

Form an quadratic equation with roots -2 ± √3.

A. x² + 4x+1=0
B.
x²-4x+1=0
C. x² - 4x + 5 =0
D.
x² + 4x + 5 =0

Answer 1

`\textit{since the roots are }-2\pm √(3)\implies \begin{cases} x=-2 + √(3)\implies&x+2-√(3)=0\\\\ x=-2 - √(3)\implies&x+2+√(3)=0 \end{cases} \\\\[-0.35em] ~\dotfill`

`( ~~ x+2-√(3) ~~ )( ~~ x+2+√(3) ~~ )=\stackrel{y}{0} \\\\\\ \underset{\textit{difference of squares}}{( ~~ \underline{(x+2)}-√(3) ~~ )( ~~ \underline{(x+2)}+√(3) ~~ )}=y\implies (x+2)^2~~ - ~~(√(3))^2~~ = ~~y \\\\\\ (\stackrel{F~O~I~L}{x^2+4x+4})~~ - ~~(3)~~ = ~~y\implies {\LARGE \begin{array}{llll} x^2+4x+1=y \end{array}}`