Solve 16*121=(80+x)(30-x)

$$16 \times 121 = ( 80 + x ) ( 30 - x )$$

Solve for x

$$x=-58$$

$$x=8$$

Steps Using the Quadratic Formula

Multiply 16 and 121 to get 1936.

$$1936=\left(80+x\right)\left(30-x\right)$$

Use the distributive property to multiply 80+x by 30-x and combine like terms.

$$1936=2400-50x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$2400-50x-x^{2}=1936$$

Subtract 1936 from both sides.

$$2400-50x-x^{2}-1936=0$$

Subtract 1936 from 2400 to get 464.

$$464-50x-x^{2}=0$$

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.

$$-x^{2}-50x+464=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -50 for b, and 464 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-1\right)\times 464}}{2\left(-1\right)}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-1\right)\times 464}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-50\right)±\sqrt{2500+4\times 464}}{2\left(-1\right)}$$

Multiply 4 times 464.

$$x=\frac{-\left(-50\right)±\sqrt{2500+1856}}{2\left(-1\right)}$$

Add 2500 to 1856.

$$x=\frac{-\left(-50\right)±\sqrt{4356}}{2\left(-1\right)}$$

Take the square root of 4356.

$$x=\frac{-\left(-50\right)±66}{2\left(-1\right)}$$

The opposite of -50 is 50.

$$x=\frac{50±66}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{50±66}{-2}$$

Now solve the equation x=\frac{50±66}{-2} when ± is plus. Add 50 to 66.

$$x=\frac{116}{-2}$$

Divide 116 by -2.

$$x=-58$$

Now solve the equation x=\frac{50±66}{-2} when ± is minus. Subtract 66 from 50.

$$x=-\frac{16}{-2}$$

Divide -16 by -2.

$$x=8$$

The equation is now solved.

$$x=-58 x=8$$

Steps for Completing the Square

Multiply 16 and 121 to get 1936.

$$1936=\left(80+x\right)\left(30-x\right)$$

Use the distributive property to multiply 80+x by 30-x and combine like terms.

$$1936=2400-50x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$2400-50x-x^{2}=1936$$

Subtract 2400 from both sides.

$$-50x-x^{2}=1936-2400$$

Subtract 2400 from 1936 to get -464.

$$-50x-x^{2}=-464$$

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.

$$-x^{2}-50x=-464$$

Divide both sides by -1.

$$\frac{-x^{2}-50x}{-1}=-\frac{464}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{50}{-1}\right)x=-\frac{464}{-1}$$

Divide -50 by -1.

$$x^{2}+50x=-\frac{464}{-1}$$

Divide -464 by -1.

$$x^{2}+50x=464$$

Divide 50, the coefficient of the x term, by 2 to get 25. Then add the square of 25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+50x+25^{2}=464+25^{2}$$

Square 25.

$$x^{2}+50x+625=464+625$$

Add 464 to 625.

$$x^{2}+50x+625=1089$$

Factor x^{2}+50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+25\right)^{2}=1089$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+25\right)^{2}}=\sqrt{1089}$$

Simplify.

$$x+25=33 x+25=-33$$

Subtract 25 from both sides of the equation.

$$x=8 x=-58$$