Solve 4x^2-12x+27=11x^2-3/5

$$4 x ^ { 2 } - 12 x + 27 = \frac { 11 x ^ { 2 } - 3 } { 5 }$$

Solve for x (complex solution)

$$x=\frac{10+\sqrt{38}i}{3}\approx 3.333333333+2.054804668i$$

$$x=\frac{-\sqrt{38}i+10}{3}\approx 3.333333333-2.054804668i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$20x^{2}-60x+135=11x^{2}-3$$

Subtract 11x^{2} from both sides.

$$20x^{2}-60x+135-11x^{2}=-3$$

Combine 20x^{2} and -11x^{2} to get 9x^{2}.

$$9x^{2}-60x+135=-3$$

Add 3 to both sides.

$$9x^{2}-60x+135+3=0$$

Add 135 and 3 to get 138.

$$9x^{2}-60x+138=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -60 for b, and 138 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 9\times 138}}{2\times 9}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 9\times 138}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-60\right)±\sqrt{3600-36\times 138}}{2\times 9}$$

Multiply -36 times 138.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4968}}{2\times 9}$$

Add 3600 to -4968.

$$x=\frac{-\left(-60\right)±\sqrt{-1368}}{2\times 9}$$

Take the square root of -1368.

$$x=\frac{-\left(-60\right)±6\sqrt{38}i}{2\times 9}$$

The opposite of -60 is 60.

$$x=\frac{60±6\sqrt{38}i}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{60±6\sqrt{38}i}{18}$$

Now solve the equation x=\frac{60±6\sqrt{38}i}{18} when ± is plus. Add 60 to 6i\sqrt{38}.

$$x=\frac{60+6\sqrt{38}i}{18}$$

Divide 60+6i\sqrt{38} by 18.

$$x=\frac{10+\sqrt{38}i}{3}$$

Now solve the equation x=\frac{60±6\sqrt{38}i}{18} when ± is minus. Subtract 6i\sqrt{38} from 60.

$$x=\frac{-6\sqrt{38}i+60}{18}$$

Divide 60-6i\sqrt{38} by 18.

$$x=\frac{-\sqrt{38}i+10}{3}$$

The equation is now solved.

$$x=\frac{10+\sqrt{38}i}{3} x=\frac{-\sqrt{38}i+10}{3}$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$20x^{2}-60x+135=11x^{2}-3$$

Subtract 11x^{2} from both sides.

$$20x^{2}-60x+135-11x^{2}=-3$$

Combine 20x^{2} and -11x^{2} to get 9x^{2}.

$$9x^{2}-60x+135=-3$$

Subtract 135 from both sides.

$$9x^{2}-60x=-3-135$$

Subtract 135 from -3 to get -138.

$$9x^{2}-60x=-138$$

Divide both sides by 9.

$$\frac{9x^{2}-60x}{9}=-\frac{138}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{60}{9}\right)x=-\frac{138}{9}$$

Reduce the fraction \frac{-60}{9} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{20}{3}x=-\frac{138}{9}$$

Reduce the fraction \frac{-138}{9} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{20}{3}x=-\frac{46}{3}$$

Divide -\frac{20}{3}, the coefficient of the x term, by 2 to get -\frac{10}{3}. Then add the square of -\frac{10}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{20}{3}x+\left(-\frac{10}{3}\right)^{2}=-\frac{46}{3}+\left(-\frac{10}{3}\right)^{2}$$

Square -\frac{10}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{20}{3}x+\frac{100}{9}=-\frac{46}{3}+\frac{100}{9}$$

Add -\frac{46}{3} to \frac{100}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{20}{3}x+\frac{100}{9}=-\frac{38}{9}$$

Factor x^{2}-\frac{20}{3}x+\frac{100}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{10}{3}\right)^{2}=-\frac{38}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{10}{3}\right)^{2}}=\sqrt{-\frac{38}{9}}$$

Simplify.

$$x-\frac{10}{3}=\frac{\sqrt{38}i}{3} x-\frac{10}{3}=-\frac{\sqrt{38}i}{3}$$

Add \frac{10}{3} to both sides of the equation.

$$x=\frac{10+\sqrt{38}i}{3} x=\frac{-\sqrt{38}i+10}{3}$$