Solve left(x+3right)^2=16(x-13)

$${ \left(x+3 \right) }^{ 2 } =16(x-13)$$

Solve for x (complex solution)

$$x=5+8\sqrt{3}i\approx 5+13.856406461i$$

$$x=-8\sqrt{3}i+5\approx 5-13.856406461i$$

Steps Using the Quadratic Formula

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

$$x^{2}+6x+9=16\left(x-13\right)$$

Use the distributive property to multiply 16 by x-13.

$$x^{2}+6x+9=16x-208$$

Subtract 16x from both sides.

$$x^{2}+6x+9-16x=-208$$

Combine 6x and -16x to get -10x.

$$x^{2}-10x+9=-208$$

Add 208 to both sides.

$$x^{2}-10x+9+208=0$$

Add 9 and 208 to get 217.

$$x^{2}-10x+217=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and 217 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 217}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 217}}{2}$$

Multiply -4 times 217.

$$x=\frac{-\left(-10\right)±\sqrt{100-868}}{2}$$

Add 100 to -868.

$$x=\frac{-\left(-10\right)±\sqrt{-768}}{2}$$

Take the square root of -768.

$$x=\frac{-\left(-10\right)±16\sqrt{3}i}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±16\sqrt{3}i}{2}$$

Now solve the equation x=\frac{10±16\sqrt{3}i}{2} when ± is plus. Add 10 to 16i\sqrt{3}.

$$x=\frac{10+16\sqrt{3}i}{2}$$

Divide 10+16i\sqrt{3} by 2.

$$x=5+8\sqrt{3}i$$

Now solve the equation x=\frac{10±16\sqrt{3}i}{2} when ± is minus. Subtract 16i\sqrt{3} from 10.

$$x=\frac{-16\sqrt{3}i+10}{2}$$

Divide 10-16i\sqrt{3} by 2.

$$x=-8\sqrt{3}i+5$$

The equation is now solved.

$$x=5+8\sqrt{3}i x=-8\sqrt{3}i+5$$

Steps for Completing the Square

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

$$x^{2}+6x+9=16\left(x-13\right)$$

Use the distributive property to multiply 16 by x-13.

$$x^{2}+6x+9=16x-208$$

Subtract 16x from both sides.

$$x^{2}+6x+9-16x=-208$$

Combine 6x and -16x to get -10x.

$$x^{2}-10x+9=-208$$

Subtract 9 from both sides.

$$x^{2}-10x=-208-9$$

Subtract 9 from -208 to get -217.

$$x^{2}-10x=-217$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-217+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-217+25$$

Add -217 to 25.

$$x^{2}-10x+25=-192$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=-192$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{-192}$$

Simplify.

$$x-5=8\sqrt{3}i x-5=-8\sqrt{3}i$$

Add 5 to both sides of the equation.

$$x=5+8\sqrt{3}i x=-8\sqrt{3}i+5$$