Solve (16+x)(40+x)=305

$$( 16 + x ) ( 40 + x ) = 305$$

Solve for x (complex solution)

$$x=\sqrt{449}-28\approx -6.8103799$$

$$x=-\left(\sqrt{449}+28\right)\approx -49.1896201$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 16+x by 40+x and combine like terms.

$$640+56x+x^{2}=305$$

Subtract 305 from both sides.

$$640+56x+x^{2}-305=0$$

Subtract 305 from 640 to get 335.

$$335+56x+x^{2}=0$$

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.

$$x^{2}+56x+335=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 56 for b, and 335 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-56±\sqrt{56^{2}-4\times 335}}{2}$$

Square 56.

$$x=\frac{-56±\sqrt{3136-4\times 335}}{2}$$

Multiply -4 times 335.

$$x=\frac{-56±\sqrt{3136-1340}}{2}$$

Add 3136 to -1340.

$$x=\frac{-56±\sqrt{1796}}{2}$$

Take the square root of 1796.

$$x=\frac{-56±2\sqrt{449}}{2}$$

Now solve the equation x=\frac{-56±2\sqrt{449}}{2} when ± is plus. Add -56 to 2\sqrt{449}.

$$x=\frac{2\sqrt{449}-56}{2}$$

Divide -56+2\sqrt{449} by 2.

$$x=\sqrt{449}-28$$

Now solve the equation x=\frac{-56±2\sqrt{449}}{2} when ± is minus. Subtract 2\sqrt{449} from -56.

$$x=\frac{-2\sqrt{449}-56}{2}$$

Divide -56-2\sqrt{449} by 2.

$$x=-\sqrt{449}-28$$

The equation is now solved.

$$x=\sqrt{449}-28 x=-\sqrt{449}-28$$

Steps for Completing the Square

Use the distributive property to multiply 16+x by 40+x and combine like terms.

$$640+56x+x^{2}=305$$

Subtract 640 from both sides.

$$56x+x^{2}=305-640$$

Subtract 640 from 305 to get -335.

$$56x+x^{2}=-335$$

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.

$$x^{2}+56x=-335$$

Divide 56, the coefficient of the x term, by 2 to get 28. Then add the square of 28 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+56x+28^{2}=-335+28^{2}$$

Square 28.

$$x^{2}+56x+784=-335+784$$

Add -335 to 784.

$$x^{2}+56x+784=449$$

Factor x^{2}+56x+784. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+28\right)^{2}=449$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+28\right)^{2}}=\sqrt{449}$$

Simplify.

$$x+28=\sqrt{449} x+28=-\sqrt{449}$$

Subtract 28 from both sides of the equation.

$$x=\sqrt{449}-28 x=-\sqrt{449}-28$$

Solve for x

$$x=\sqrt{449}-28\approx -6.8103799$$

$$x=-\sqrt{449}-28\approx -49.1896201$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 16+x by 40+x and combine like terms.

$$640+56x+x^{2}=305$$

Subtract 305 from both sides.

$$640+56x+x^{2}-305=0$$

Subtract 305 from 640 to get 335.

$$335+56x+x^{2}=0$$

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.

$$x^{2}+56x+335=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 56 for b, and 335 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-56±\sqrt{56^{2}-4\times 335}}{2}$$

Square 56.

$$x=\frac{-56±\sqrt{3136-4\times 335}}{2}$$

Multiply -4 times 335.

$$x=\frac{-56±\sqrt{3136-1340}}{2}$$

Add 3136 to -1340.

$$x=\frac{-56±\sqrt{1796}}{2}$$

Take the square root of 1796.

$$x=\frac{-56±2\sqrt{449}}{2}$$

Now solve the equation x=\frac{-56±2\sqrt{449}}{2} when ± is plus. Add -56 to 2\sqrt{449}.

$$x=\frac{2\sqrt{449}-56}{2}$$

Divide -56+2\sqrt{449} by 2.

$$x=\sqrt{449}-28$$

Now solve the equation x=\frac{-56±2\sqrt{449}}{2} when ± is minus. Subtract 2\sqrt{449} from -56.

$$x=\frac{-2\sqrt{449}-56}{2}$$

Divide -56-2\sqrt{449} by 2.

$$x=-\sqrt{449}-28$$

The equation is now solved.

$$x=\sqrt{449}-28 x=-\sqrt{449}-28$$

Steps for Completing the Square

Use the distributive property to multiply 16+x by 40+x and combine like terms.

$$640+56x+x^{2}=305$$

Subtract 640 from both sides.

$$56x+x^{2}=305-640$$

Subtract 640 from 305 to get -335.

$$56x+x^{2}=-335$$

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.

$$x^{2}+56x=-335$$

Divide 56, the coefficient of the x term, by 2 to get 28. Then add the square of 28 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+56x+28^{2}=-335+28^{2}$$

Square 28.

$$x^{2}+56x+784=-335+784$$

Add -335 to 784.

$$x^{2}+56x+784=449$$

Factor x^{2}+56x+784. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+28\right)^{2}=449$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+28\right)^{2}}=\sqrt{449}$$

Simplify.

$$x+28=\sqrt{449} x+28=-\sqrt{449}$$

Subtract 28 from both sides of the equation.

$$x=\sqrt{449}-28 x=-\sqrt{449}-28$$