```Question 191307
a)

Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-6}}}, and {{{c=9}}}

Let's use the quadratic formula to solve for x

{{{x = (-(-6) +- sqrt( (-6)^2-4(1)(9) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-6}}}, and {{{c=9}}}

{{{x = (6 +- sqrt( (-6)^2-4(1)(9) ))/(2(1))}}} Negate {{{-6}}} to get {{{6}}}.

{{{x = (6 +- sqrt( 36-4(1)(9) ))/(2(1))}}} Square {{{-6}}} to get {{{36}}}.

{{{x = (6 +- sqrt( 36-36 ))/(2(1))}}} Multiply {{{4(1)(9)}}} to get {{{36}}}

{{{x = (6 +- sqrt( 0 ))/(2(1))}}} Subtract {{{36}}} from {{{36}}} to get {{{0}}}

{{{x = (6 +- sqrt( 0 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}.

{{{x = (6 +- 0)/(2)}}} Take the square root of {{{0}}} to get {{{0}}}.

{{{x = (6 + 0)/(2)}}} or {{{x = (6 - 0)/(2)}}} Break up the expression.

{{{x = (6)/(2)}}} or {{{x =  (6)/(2)}}} Combine like terms.

{{{x = 3}}} or {{{x = 3}}} Simplify.

So solution is {{{x = 3}}}

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b)

Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=4}}}, and {{{c=7}}}

Let's use the quadratic formula to solve for x

{{{x = (-(4) +- sqrt( (4)^2-4(1)(7) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=4}}}, and {{{c=7}}}

{{{x = (-4 +- sqrt( 16-4(1)(7) ))/(2(1))}}} Square {{{4}}} to get {{{16}}}.

{{{x = (-4 +- sqrt( 16-28 ))/(2(1))}}} Multiply {{{4(1)(7)}}} to get {{{28}}}

{{{x = (-4 +- sqrt( -12 ))/(2(1))}}} Subtract {{{28}}} from {{{16}}} to get {{{-12}}}

{{{x = (-4 +- sqrt( -12 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}.

{{{x = (-4 +- 2i*sqrt(3))/(2)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)

{{{x = (-4)/(2) +- (2i*sqrt(3))/(2)}}} Break up the fraction.

{{{x = -2 +- sqrt(3)*i}}} Reduce.

{{{x = -2+sqrt(3)*i}}} or {{{x = -2-sqrt(3)*i}}} Break up the expression.

So the answers are {{{x = -2+sqrt(3)*i}}} or {{{x = -2-sqrt(3)*i}}} where {{{i=sqrt(-1)}}}

Note: the solutions are in the form {{{x=a+bi}}} or {{{x=a-bi}}} where {{{a=-2}}} and {{{b=sqrt(3)}}}```