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

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{47}i-17}{2}\approx -8.5-3.4278273i$$

$$x=\frac{-17+\sqrt{47}i}{2}\approx -8.5+3.4278273i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$x=x^{2}+18x+81+3$$

Add 81 and 3 to get 84.

$$x=x^{2}+18x+84$$

Subtract x^{2} from both sides.

$$x-x^{2}=18x+84$$

Subtract 18x from both sides.

$$x-x^{2}-18x=84$$

Combine x and -18x to get -17x.

$$-17x-x^{2}=84$$

Subtract 84 from both sides.

$$-17x-x^{2}-84=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-17x-84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -17 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\left(-1\right)\left(-84\right)}}{2\left(-1\right)}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\left(-1\right)\left(-84\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-17\right)±\sqrt{289+4\left(-84\right)}}{2\left(-1\right)}$$

Multiply 4 times -84.

$$x=\frac{-\left(-17\right)±\sqrt{289-336}}{2\left(-1\right)}$$

Add 289 to -336.

$$x=\frac{-\left(-17\right)±\sqrt{-47}}{2\left(-1\right)}$$

Take the square root of -47.

$$x=\frac{-\left(-17\right)±\sqrt{47}i}{2\left(-1\right)}$$

The opposite of -17 is 17.

$$x=\frac{17±\sqrt{47}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{17±\sqrt{47}i}{-2}$$

Now solve the equation x=\frac{17±\sqrt{47}i}{-2} when ± is plus. Add 17 to i\sqrt{47}.

$$x=\frac{17+\sqrt{47}i}{-2}$$

Divide 17+i\sqrt{47} by -2.

$$x=\frac{-\sqrt{47}i-17}{2}$$

Now solve the equation x=\frac{17±\sqrt{47}i}{-2} when ± is minus. Subtract i\sqrt{47} from 17.

$$x=\frac{-\sqrt{47}i+17}{-2}$$

Divide 17-i\sqrt{47} by -2.

$$x=\frac{-17+\sqrt{47}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{47}i-17}{2} x=\frac{-17+\sqrt{47}i}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$x=x^{2}+18x+81+3$$

Add 81 and 3 to get 84.

$$x=x^{2}+18x+84$$

Subtract x^{2} from both sides.

$$x-x^{2}=18x+84$$

Subtract 18x from both sides.

$$x-x^{2}-18x=84$$

Combine x and -18x to get -17x.

$$-17x-x^{2}=84$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}-17x=84$$

Divide both sides by -1.

$$\frac{-x^{2}-17x}{-1}=\frac{84}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{17}{-1}\right)x=\frac{84}{-1}$$

Divide -17 by -1.

$$x^{2}+17x=\frac{84}{-1}$$

Divide 84 by -1.

$$x^{2}+17x=-84$$

Divide 17, the coefficient of the x term, by 2 to get \frac{17}{2}. Then add the square of \frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+17x+\left(\frac{17}{2}\right)^{2}=-84+\left(\frac{17}{2}\right)^{2}$$

Square \frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+17x+\frac{289}{4}=-84+\frac{289}{4}$$

Add -84 to \frac{289}{4}.

$$x^{2}+17x+\frac{289}{4}=-\frac{47}{4}$$

Factor x^{2}+17x+\frac{289}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{17}{2}\right)^{2}=-\frac{47}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{2}\right)^{2}}=\sqrt{-\frac{47}{4}}$$

Simplify.

$$x+\frac{17}{2}=\frac{\sqrt{47}i}{2} x+\frac{17}{2}=-\frac{\sqrt{47}i}{2}$$

Subtract \frac{17}{2} from both sides of the equation.

$$x=\frac{-17+\sqrt{47}i}{2} x=\frac{-\sqrt{47}i-17}{2}$$