Solve 2(3x+4/5)^2/3=4-7

$$\frac { 2 ( \frac { 3 x + 4 } { 5 } ) ^ { 2 } } { 3 } = 4 - 7$$

Solve for x (complex solution)

$$x=-\frac{5\sqrt{2}i}{2}-\frac{4}{3}\approx -1.333333333-3.535533906i$$

$$x=\frac{5\sqrt{2}i}{2}-\frac{4}{3}\approx -1.333333333+3.535533906i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$2\times \left(\frac{3x+4}{5}\right)^{2}=12-21$$

To raise \frac{3x+4}{5} to a power, raise both numerator and denominator to the power and then divide.

$$2\times \frac{\left(3x+4\right)^{2}}{5^{2}}=12-21$$

Express 2\times \frac{\left(3x+4\right)^{2}}{5^{2}} as a single fraction.

$$\frac{2\left(3x+4\right)^{2}}{5^{2}}=12-21$$

Subtract 21 from 12 to get -9.

$$\frac{2\left(3x+4\right)^{2}}{5^{2}}=-9$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+4\right)^{2}.

$$\frac{2\left(9x^{2}+24x+16\right)}{5^{2}}=-9$$

Calculate 5 to the power of 2 and get 25.

$$\frac{2\left(9x^{2}+24x+16\right)}{25}=-9$$

Add 9 to both sides.

$$\frac{2\left(9x^{2}+24x+16\right)}{25}+9=0$$

Use the distributive property to multiply 2 by 9x^{2}+24x+16.

$$\frac{18x^{2}+48x+32}{25}+9=0$$

Divide each term of 18x^{2}+48x+32 by 25 to get \frac{18}{25}x^{2}+\frac{48}{25}x+\frac{32}{25}.

$$\frac{18}{25}x^{2}+\frac{48}{25}x+\frac{32}{25}+9=0$$

Add \frac{32}{25} and 9 to get \frac{257}{25}.

$$\frac{18}{25}x^{2}+\frac{48}{25}x+\frac{257}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{18}{25} for a, \frac{48}{25} for b, and \frac{257}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{48}{25}±\sqrt{\left(\frac{48}{25}\right)^{2}-4\times \frac{18}{25}\times \frac{257}{25}}}{2\times \frac{18}{25}}$$

Square \frac{48}{25} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{48}{25}±\sqrt{\frac{2304}{625}-4\times \frac{18}{25}\times \frac{257}{25}}}{2\times \frac{18}{25}}$$

Multiply -4 times \frac{18}{25}.

$$x=\frac{-\frac{48}{25}±\sqrt{\frac{2304}{625}-\frac{72}{25}\times \frac{257}{25}}}{2\times \frac{18}{25}}$$

Multiply -\frac{72}{25} times \frac{257}{25} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{48}{25}±\sqrt{\frac{2304-18504}{625}}}{2\times \frac{18}{25}}$$

Add \frac{2304}{625} to -\frac{18504}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{48}{25}±\sqrt{-\frac{648}{25}}}{2\times \frac{18}{25}}$$

Take the square root of -\frac{648}{25}.

$$x=\frac{-\frac{48}{25}±\frac{18\sqrt{2}i}{5}}{2\times \frac{18}{25}}$$

Multiply 2 times \frac{18}{25}.

$$x=\frac{-\frac{48}{25}±\frac{18\sqrt{2}i}{5}}{\frac{36}{25}}$$

Now solve the equation x=\frac{-\frac{48}{25}±\frac{18\sqrt{2}i}{5}}{\frac{36}{25}} when ± is plus. Add -\frac{48}{25} to \frac{18i\sqrt{2}}{5}.

$$x=\frac{\frac{18\sqrt{2}i}{5}-\frac{48}{25}}{\frac{36}{25}}$$

Divide -\frac{48}{25}+\frac{18i\sqrt{2}}{5} by \frac{36}{25} by multiplying -\frac{48}{25}+\frac{18i\sqrt{2}}{5} by the reciprocal of \frac{36}{25}.

$$x=\frac{5\sqrt{2}i}{2}-\frac{4}{3}$$

Now solve the equation x=\frac{-\frac{48}{25}±\frac{18\sqrt{2}i}{5}}{\frac{36}{25}} when ± is minus. Subtract \frac{18i\sqrt{2}}{5} from -\frac{48}{25}.

$$x=\frac{-\frac{18\sqrt{2}i}{5}-\frac{48}{25}}{\frac{36}{25}}$$

Divide -\frac{48}{25}-\frac{18i\sqrt{2}}{5} by \frac{36}{25} by multiplying -\frac{48}{25}-\frac{18i\sqrt{2}}{5} by the reciprocal of \frac{36}{25}.

$$x=-\frac{5\sqrt{2}i}{2}-\frac{4}{3}$$

The equation is now solved.

$$x=\frac{5\sqrt{2}i}{2}-\frac{4}{3} x=-\frac{5\sqrt{2}i}{2}-\frac{4}{3}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$2\times \left(\frac{3x+4}{5}\right)^{2}=12-21$$

To raise \frac{3x+4}{5} to a power, raise both numerator and denominator to the power and then divide.

$$2\times \frac{\left(3x+4\right)^{2}}{5^{2}}=12-21$$

Express 2\times \frac{\left(3x+4\right)^{2}}{5^{2}} as a single fraction.

$$\frac{2\left(3x+4\right)^{2}}{5^{2}}=12-21$$

Subtract 21 from 12 to get -9.

$$\frac{2\left(3x+4\right)^{2}}{5^{2}}=-9$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+4\right)^{2}.

$$\frac{2\left(9x^{2}+24x+16\right)}{5^{2}}=-9$$

Calculate 5 to the power of 2 and get 25.

$$\frac{2\left(9x^{2}+24x+16\right)}{25}=-9$$

Multiply both sides by 25.

$$2\left(9x^{2}+24x+16\right)=-9\times 25$$

Use the distributive property to multiply 2 by 9x^{2}+24x+16.

$$18x^{2}+48x+32=-9\times 25$$

Multiply -9 and 25 to get -225.

$$18x^{2}+48x+32=-225$$

Subtract 32 from both sides.

$$18x^{2}+48x=-225-32$$

Subtract 32 from -225 to get -257.

$$18x^{2}+48x=-257$$

Divide both sides by 18.

$$\frac{18x^{2}+48x}{18}=-\frac{257}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x^{2}+\frac{48}{18}x=-\frac{257}{18}$$

Reduce the fraction \frac{48}{18} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{8}{3}x=-\frac{257}{18}$$

Divide \frac{8}{3}, the coefficient of the x term, by 2 to get \frac{4}{3}. Then add the square of \frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8}{3}x+\left(\frac{4}{3}\right)^{2}=-\frac{257}{18}+\left(\frac{4}{3}\right)^{2}$$

Square \frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=-\frac{257}{18}+\frac{16}{9}$$

Add -\frac{257}{18} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=-\frac{25}{2}$$

Factor x^{2}+\frac{8}{3}x+\frac{16}{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}.

$$\left(x+\frac{4}{3}\right)^{2}=-\frac{25}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4}{3}\right)^{2}}=\sqrt{-\frac{25}{2}}$$

Simplify.

$$x+\frac{4}{3}=\frac{5\sqrt{2}i}{2} x+\frac{4}{3}=-\frac{5\sqrt{2}i}{2}$$

Subtract \frac{4}{3} from both sides of the equation.

$$x=\frac{5\sqrt{2}i}{2}-\frac{4}{3} x=-\frac{5\sqrt{2}i}{2}-\frac{4}{3}$$