Solve (2x+3)^2=-9(x-1)^2

$$( 2 x + 3 ) ^ { 2 } = - 9 ( x - 1 ) ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{3}{13}+\frac{15}{13}i\approx 0.230769231+1.153846154i$$

$$x=\frac{3}{13}-\frac{15}{13}i\approx 0.230769231-1.153846154i$$

Steps Using the Quadratic Formula

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

$$4x^{2}+12x+9=-9\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$4x^{2}+12x+9=-9\left(x^{2}-2x+1\right)$$

Use the distributive property to multiply -9 by x^{2}-2x+1.

$$4x^{2}+12x+9=-9x^{2}+18x-9$$

Add 9x^{2} to both sides.

$$4x^{2}+12x+9+9x^{2}=18x-9$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$13x^{2}+12x+9=18x-9$$

Subtract 18x from both sides.

$$13x^{2}+12x+9-18x=-9$$

Combine 12x and -18x to get -6x.

$$13x^{2}-6x+9=-9$$

Add 9 to both sides.

$$13x^{2}-6x+9+9=0$$

Add 9 and 9 to get 18.

$$13x^{2}-6x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, -6 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 13\times 18}}{2\times 13}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 13\times 18}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-\left(-6\right)±\sqrt{36-52\times 18}}{2\times 13}$$

Multiply -52 times 18.

$$x=\frac{-\left(-6\right)±\sqrt{36-936}}{2\times 13}$$

Add 36 to -936.

$$x=\frac{-\left(-6\right)±\sqrt{-900}}{2\times 13}$$

Take the square root of -900.

$$x=\frac{-\left(-6\right)±30i}{2\times 13}$$

The opposite of -6 is 6.

$$x=\frac{6±30i}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{6±30i}{26}$$

Now solve the equation x=\frac{6±30i}{26} when ± is plus. Add 6 to 30i.

$$x=\frac{6+30i}{26}$$

Divide 6+30i by 26.

$$x=\frac{3}{13}+\frac{15}{13}i$$

Now solve the equation x=\frac{6±30i}{26} when ± is minus. Subtract 30i from 6.

$$x=\frac{6-30i}{26}$$

Divide 6-30i by 26.

$$x=\frac{3}{13}-\frac{15}{13}i$$

The equation is now solved.

$$x=\frac{3}{13}+\frac{15}{13}i x=\frac{3}{13}-\frac{15}{13}i$$

Steps for Completing the Square

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

$$4x^{2}+12x+9=-9\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$4x^{2}+12x+9=-9\left(x^{2}-2x+1\right)$$

Use the distributive property to multiply -9 by x^{2}-2x+1.

$$4x^{2}+12x+9=-9x^{2}+18x-9$$

Add 9x^{2} to both sides.

$$4x^{2}+12x+9+9x^{2}=18x-9$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$13x^{2}+12x+9=18x-9$$

Subtract 18x from both sides.

$$13x^{2}+12x+9-18x=-9$$

Combine 12x and -18x to get -6x.

$$13x^{2}-6x+9=-9$$

Subtract 9 from both sides.

$$13x^{2}-6x=-9-9$$

Subtract 9 from -9 to get -18.

$$13x^{2}-6x=-18$$

Divide both sides by 13.

$$\frac{13x^{2}-6x}{13}=-\frac{18}{13}$$

Dividing by 13 undoes the multiplication by 13.

$$x^{2}-\frac{6}{13}x=-\frac{18}{13}$$

Divide -\frac{6}{13}, the coefficient of the x term, by 2 to get -\frac{3}{13}. Then add the square of -\frac{3}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6}{13}x+\left(-\frac{3}{13}\right)^{2}=-\frac{18}{13}+\left(-\frac{3}{13}\right)^{2}$$

Square -\frac{3}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6}{13}x+\frac{9}{169}=-\frac{18}{13}+\frac{9}{169}$$

Add -\frac{18}{13} to \frac{9}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6}{13}x+\frac{9}{169}=-\frac{225}{169}$$

Factor x^{2}-\frac{6}{13}x+\frac{9}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{13}\right)^{2}=-\frac{225}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{13}\right)^{2}}=\sqrt{-\frac{225}{169}}$$

Simplify.

$$x-\frac{3}{13}=\frac{15}{13}i x-\frac{3}{13}=-\frac{15}{13}i$$

Add \frac{3}{13} to both sides of the equation.

$$x=\frac{3}{13}+\frac{15}{13}i x=\frac{3}{13}-\frac{15}{13}i$$