Solve -(3+h)^2-4(1)(3h+1)=0

$$- ( 3 + h ) ^ { 2 } - 4 ( 1 ) ( 3 h + 1 ) = 0$$

Solve for h

$$h=2\sqrt{17}-9\approx -0.753788749$$

$$h=-2\sqrt{17}-9\approx -17.246211251$$

Steps Using the Quadratic Formula

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

$$-\left(9+6h+h^{2}\right)-4\times 1\left(3h+1\right)=0$$

To find the opposite of 9+6h+h^{2}, find the opposite of each term.

$$-9-6h-h^{2}-4\times 1\left(3h+1\right)=0$$

Multiply 4 and 1 to get 4.

$$-9-6h-h^{2}-4\left(3h+1\right)=0$$

Use the distributive property to multiply -4 by 3h+1.

$$-9-6h-h^{2}-12h-4=0$$

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

$$-9-18h-h^{2}-4=0$$

Subtract 4 from -9 to get -13.

$$-13-18h-h^{2}=0$$

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.

$$-h^{2}-18h-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -18 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-1\right)\left(-13\right)}}{2\left(-1\right)}$$

Square -18.

$$h=\frac{-\left(-18\right)±\sqrt{324-4\left(-1\right)\left(-13\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$h=\frac{-\left(-18\right)±\sqrt{324+4\left(-13\right)}}{2\left(-1\right)}$$

Multiply 4 times -13.

$$h=\frac{-\left(-18\right)±\sqrt{324-52}}{2\left(-1\right)}$$

Add 324 to -52.

$$h=\frac{-\left(-18\right)±\sqrt{272}}{2\left(-1\right)}$$

Take the square root of 272.

$$h=\frac{-\left(-18\right)±4\sqrt{17}}{2\left(-1\right)}$$

The opposite of -18 is 18.

$$h=\frac{18±4\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$h=\frac{18±4\sqrt{17}}{-2}$$

Now solve the equation h=\frac{18±4\sqrt{17}}{-2} when ± is plus. Add 18 to 4\sqrt{17}.

$$h=\frac{4\sqrt{17}+18}{-2}$$

Divide 18+4\sqrt{17} by -2.

$$h=-2\sqrt{17}-9$$

Now solve the equation h=\frac{18±4\sqrt{17}}{-2} when ± is minus. Subtract 4\sqrt{17} from 18.

$$h=\frac{18-4\sqrt{17}}{-2}$$

Divide 18-4\sqrt{17} by -2.

$$h=2\sqrt{17}-9$$

The equation is now solved.

$$h=-2\sqrt{17}-9 h=2\sqrt{17}-9$$

Steps for Completing the Square

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

$$-\left(9+6h+h^{2}\right)-4\times 1\left(3h+1\right)=0$$

To find the opposite of 9+6h+h^{2}, find the opposite of each term.

$$-9-6h-h^{2}-4\times 1\left(3h+1\right)=0$$

Multiply 4 and 1 to get 4.

$$-9-6h-h^{2}-4\left(3h+1\right)=0$$

Use the distributive property to multiply -4 by 3h+1.

$$-9-6h-h^{2}-12h-4=0$$

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

$$-9-18h-h^{2}-4=0$$

Subtract 4 from -9 to get -13.

$$-13-18h-h^{2}=0$$

Add 13 to both sides. Anything plus zero gives itself.

$$-18h-h^{2}=13$$

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.

$$-h^{2}-18h=13$$

Divide both sides by -1.

$$\frac{-h^{2}-18h}{-1}=\frac{13}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$h^{2}+\left(-\frac{18}{-1}\right)h=\frac{13}{-1}$$

Divide -18 by -1.

$$h^{2}+18h=\frac{13}{-1}$$

Divide 13 by -1.

$$h^{2}+18h=-13$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}+18h+9^{2}=-13+9^{2}$$

Square 9.

$$h^{2}+18h+81=-13+81$$

Add -13 to 81.

$$h^{2}+18h+81=68$$

Factor h^{2}+18h+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h+9\right)^{2}=68$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+9\right)^{2}}=\sqrt{68}$$

Simplify.

$$h+9=2\sqrt{17} h+9=-2\sqrt{17}$$

Subtract 9 from both sides of the equation.

$$h=2\sqrt{17}-9 h=-2\sqrt{17}-9$$