Solve x+1=(x+7)^2

$$x + 1 = ( x + 7 ) ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{23}i-13}{2}\approx -6.5-2.397915762i$$

$$x=\frac{-13+\sqrt{23}i}{2}\approx -6.5+2.397915762i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x+1=x^{2}+14x+49$$

Subtract x^{2} from both sides.

$$x+1-x^{2}=14x+49$$

Subtract 14x from both sides.

$$x+1-x^{2}-14x=49$$

Combine x and -14x to get -13x.

$$-13x+1-x^{2}=49$$

Subtract 49 from both sides.

$$-13x+1-x^{2}-49=0$$

Subtract 49 from 1 to get -48.

$$-13x-48-x^{2}=0$$

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.

$$-x^{2}-13x-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -13 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-1\right)\left(-48\right)}}{2\left(-1\right)}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-1\right)\left(-48\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-13\right)±\sqrt{169+4\left(-48\right)}}{2\left(-1\right)}$$

Multiply 4 times -48.

$$x=\frac{-\left(-13\right)±\sqrt{169-192}}{2\left(-1\right)}$$

Add 169 to -192.

$$x=\frac{-\left(-13\right)±\sqrt{-23}}{2\left(-1\right)}$$

Take the square root of -23.

$$x=\frac{-\left(-13\right)±\sqrt{23}i}{2\left(-1\right)}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{23}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{13±\sqrt{23}i}{-2}$$

Now solve the equation x=\frac{13±\sqrt{23}i}{-2} when ± is plus. Add 13 to i\sqrt{23}.

$$x=\frac{13+\sqrt{23}i}{-2}$$

Divide 13+i\sqrt{23} by -2.

$$x=\frac{-\sqrt{23}i-13}{2}$$

Now solve the equation x=\frac{13±\sqrt{23}i}{-2} when ± is minus. Subtract i\sqrt{23} from 13.

$$x=\frac{-\sqrt{23}i+13}{-2}$$

Divide 13-i\sqrt{23} by -2.

$$x=\frac{-13+\sqrt{23}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{23}i-13}{2} x=\frac{-13+\sqrt{23}i}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x+1=x^{2}+14x+49$$

Subtract x^{2} from both sides.

$$x+1-x^{2}=14x+49$$

Subtract 14x from both sides.

$$x+1-x^{2}-14x=49$$

Combine x and -14x to get -13x.

$$-13x+1-x^{2}=49$$

Subtract 1 from both sides.

$$-13x-x^{2}=49-1$$

Subtract 1 from 49 to get 48.

$$-13x-x^{2}=48$$

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.

$$-x^{2}-13x=48$$

Divide both sides by -1.

$$\frac{-x^{2}-13x}{-1}=\frac{48}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{13}{-1}\right)x=\frac{48}{-1}$$

Divide -13 by -1.

$$x^{2}+13x=\frac{48}{-1}$$

Divide 48 by -1.

$$x^{2}+13x=-48$$

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

$$x^{2}+13x+\left(\frac{13}{2}\right)^{2}=-48+\left(\frac{13}{2}\right)^{2}$$

Square \frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+13x+\frac{169}{4}=-48+\frac{169}{4}$$

Add -48 to \frac{169}{4}.

$$x^{2}+13x+\frac{169}{4}=-\frac{23}{4}$$

Factor x^{2}+13x+\frac{169}{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}.

$$\left(x+\frac{13}{2}\right)^{2}=-\frac{23}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Simplify.

$$x+\frac{13}{2}=\frac{\sqrt{23}i}{2} x+\frac{13}{2}=-\frac{\sqrt{23}i}{2}$$

Subtract \frac{13}{2} from both sides of the equation.

$$x=\frac{-13+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i-13}{2}$$