How do you find the roots, real and imaginary, of y= x^2-6x+(x-4)^2  using the quadratic formula?

Mar 16, 2016

real roots: $x = \frac{7 \pm \sqrt{17}}{2}$
imaginary roorts: none

Explanation:

$1$ Start by converting the equation into standard form.

$y = {x}^{2} - 6 x + {\left(x - 4\right)}^{2}$

y=x^2-6x+(color(crimson)x color(blue)(-4))(color(orange)x color(teal)(-4))

y=x^2-6x+(color(crimson)x(color(orange)x) $\textcolor{c r i m s o n}{+ x} \left(\textcolor{t e a l}{- 4}\right)$ $\textcolor{b l u e}{- 4} \left(\textcolor{\mathmr{and} a n \ge}{x}\right)$ color(blue)(-4)(color(teal)(-4)))

$y = {x}^{2} - 6 x + \left({x}^{2} - 4 x - 4 x + 16\right)$

$y = {x}^{2} - 6 x + \left({x}^{2} - 8 x + 16\right)$

$y = {x}^{2} + {x}^{2} - 6 x - 8 x + 16$

$y = \textcolor{b r o w n}{2} {x}^{2}$ $\textcolor{t u r q u o i s e}{- 14} x$ $\textcolor{v i o \le t}{+ 16}$

$2$. Identify the $\textcolor{b r o w n}{a} , \textcolor{t u r q u o i s e}{b} ,$ and $\textcolor{v i o \le t}{c}$ values of the quadratic equation. Then plug the values into the quadratic formula to solve for the roots.

$\textcolor{b r o w n}{a = 2} \textcolor{w h i t e}{X X X X X} \textcolor{t u r q u o i s e}{b = - 14} \textcolor{w h i t e}{X X X X X} \textcolor{v i o \le t}{c = 16}$

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$x = \frac{- \left(\textcolor{t u r q u o i s e}{- 14}\right) \pm \sqrt{{\left(\textcolor{t u r q u o i s e}{- 14}\right)}^{2} - 4 \left(\textcolor{b r o w n}{2}\right) \left(\textcolor{v i o \le t}{16}\right)}}{2 \left(\textcolor{b r o w n}{2}\right)}$

$x = \frac{14 \pm \sqrt{196 - 128}}{4}$

$x = \frac{14 \pm \sqrt{68}}{4}$

$x = \frac{14 \pm 2 \sqrt{17}}{4}$

$x = \frac{2 \left(7 \pm \sqrt{17}\right)}{2 \left(2\right)}$

$x = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \left(7 \pm \sqrt{17}\right)}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \left(2\right)}$

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} x = \frac{7 \pm \sqrt{17}}{2} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\therefore$, the real roots are $x = \frac{7 \pm \sqrt{17}}{2}$.