Solve left(68+2dright)(68+d)=144

$$\left( 68+2d \right) (68+d) = 144$$

Solve for d

$$d=-70$$

$$d=-32$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 68+2d by 68+d and combine like terms.

$$4624+204d+2d^{2}=144$$

Subtract 144 from both sides.

$$4624+204d+2d^{2}-144=0$$

Subtract 144 from 4624 to get 4480.

$$4480+204d+2d^{2}=0$$

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.

$$2d^{2}+204d+4480=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 204 for b, and 4480 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-204±\sqrt{204^{2}-4\times 2\times 4480}}{2\times 2}$$

Square 204.

$$d=\frac{-204±\sqrt{41616-4\times 2\times 4480}}{2\times 2}$$

Multiply -4 times 2.

$$d=\frac{-204±\sqrt{41616-8\times 4480}}{2\times 2}$$

Multiply -8 times 4480.

$$d=\frac{-204±\sqrt{41616-35840}}{2\times 2}$$

Add 41616 to -35840.

$$d=\frac{-204±\sqrt{5776}}{2\times 2}$$

Take the square root of 5776.

$$d=\frac{-204±76}{2\times 2}$$

Multiply 2 times 2.

$$d=\frac{-204±76}{4}$$

Now solve the equation d=\frac{-204±76}{4} when ± is plus. Add -204 to 76.

$$d=-\frac{128}{4}$$

Divide -128 by 4.

$$d=-32$$

Now solve the equation d=\frac{-204±76}{4} when ± is minus. Subtract 76 from -204.

$$d=-\frac{280}{4}$$

Divide -280 by 4.

$$d=-70$$

The equation is now solved.

$$d=-32 d=-70$$

Steps for Completing the Square

Use the distributive property to multiply 68+2d by 68+d and combine like terms.

$$4624+204d+2d^{2}=144$$

Subtract 4624 from both sides.

$$204d+2d^{2}=144-4624$$

Subtract 4624 from 144 to get -4480.

$$204d+2d^{2}=-4480$$

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.

$$2d^{2}+204d=-4480$$

Divide both sides by 2.

$$\frac{2d^{2}+204d}{2}=-\frac{4480}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$d^{2}+\frac{204}{2}d=-\frac{4480}{2}$$

Divide 204 by 2.

$$d^{2}+102d=-\frac{4480}{2}$$

Divide -4480 by 2.

$$d^{2}+102d=-2240$$

Divide 102, the coefficient of the x term, by 2 to get 51. Then add the square of 51 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}+102d+51^{2}=-2240+51^{2}$$

Square 51.

$$d^{2}+102d+2601=-2240+2601$$

Add -2240 to 2601.

$$d^{2}+102d+2601=361$$

Factor d^{2}+102d+2601. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d+51\right)^{2}=361$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+51\right)^{2}}=\sqrt{361}$$

Simplify.

$$d+51=19 d+51=-19$$

Subtract 51 from both sides of the equation.

$$d=-32 d=-70$$