Solve 16*121=(80+x)*(10-x)

$$16 \times 121 = ( 80 + x ) \times ( 10 - x )$$

Solve for x (complex solution)

$$x=\sqrt{89}-35\approx -25.566018868$$

$$x=-\left(\sqrt{89}+35\right)\approx -44.433981132$$

Steps Using the Quadratic Formula

Multiply 16 and 121 to get 1936.

$$1936=\left(80+x\right)\left(10-x\right)$$

Use the distributive property to multiply 80+x by 10-x and combine like terms.

$$1936=800-70x-x^{2}$$

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

$$800-70x-x^{2}=1936$$

Subtract 1936 from both sides.

$$800-70x-x^{2}-1936=0$$

Subtract 1936 from 800 to get -1136.

$$-1136-70x-x^{2}=0$$

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.

$$-x^{2}-70x-1136=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -70 for b, and -1136 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\left(-1\right)\left(-1136\right)}}{2\left(-1\right)}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\left(-1\right)\left(-1136\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-70\right)±\sqrt{4900+4\left(-1136\right)}}{2\left(-1\right)}$$

Multiply 4 times -1136.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4544}}{2\left(-1\right)}$$

Add 4900 to -4544.

$$x=\frac{-\left(-70\right)±\sqrt{356}}{2\left(-1\right)}$$

Take the square root of 356.

$$x=\frac{-\left(-70\right)±2\sqrt{89}}{2\left(-1\right)}$$

The opposite of -70 is 70.

$$x=\frac{70±2\sqrt{89}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{70±2\sqrt{89}}{-2}$$

Now solve the equation x=\frac{70±2\sqrt{89}}{-2} when ± is plus. Add 70 to 2\sqrt{89}.

$$x=\frac{2\sqrt{89}+70}{-2}$$

Divide 70+2\sqrt{89} by -2.

$$x=-\left(\sqrt{89}+35\right)$$

Now solve the equation x=\frac{70±2\sqrt{89}}{-2} when ± is minus. Subtract 2\sqrt{89} from 70.

$$x=\frac{70-2\sqrt{89}}{-2}$$

Divide 70-2\sqrt{89} by -2.

$$x=\sqrt{89}-35$$

The equation is now solved.

$$x=-\left(\sqrt{89}+35\right) x=\sqrt{89}-35$$

Steps for Completing the Square

Multiply 16 and 121 to get 1936.

$$1936=\left(80+x\right)\left(10-x\right)$$

Use the distributive property to multiply 80+x by 10-x and combine like terms.

$$1936=800-70x-x^{2}$$

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

$$800-70x-x^{2}=1936$$

Subtract 800 from both sides.

$$-70x-x^{2}=1936-800$$

Subtract 800 from 1936 to get 1136.

$$-70x-x^{2}=1136$$

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.

$$-x^{2}-70x=1136$$

Divide both sides by -1.

$$\frac{-x^{2}-70x}{-1}=\frac{1136}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{70}{-1}\right)x=\frac{1136}{-1}$$

Divide -70 by -1.

$$x^{2}+70x=\frac{1136}{-1}$$

Divide 1136 by -1.

$$x^{2}+70x=-1136$$

Divide 70, the coefficient of the x term, by 2 to get 35. Then add the square of 35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+70x+35^{2}=-1136+35^{2}$$

Square 35.

$$x^{2}+70x+1225=-1136+1225$$

Add -1136 to 1225.

$$x^{2}+70x+1225=89$$

Factor x^{2}+70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+35\right)^{2}=89$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+35\right)^{2}}=\sqrt{89}$$

Simplify.

$$x+35=\sqrt{89} x+35=-\sqrt{89}$$

Subtract 35 from both sides of the equation.

$$x=\sqrt{89}-35 x=-\sqrt{89}-35$$

Solve for x

$$x=\sqrt{89}-35\approx -25.566018868$$

$$x=-\sqrt{89}-35\approx -44.433981132$$

Steps Using the Quadratic Formula

Multiply 16 and 121 to get 1936.

$$1936=\left(80+x\right)\left(10-x\right)$$

Use the distributive property to multiply 80+x by 10-x and combine like terms.

$$1936=800-70x-x^{2}$$

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

$$800-70x-x^{2}=1936$$

Subtract 1936 from both sides.

$$800-70x-x^{2}-1936=0$$

Subtract 1936 from 800 to get -1136.

$$-1136-70x-x^{2}=0$$

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.

$$-x^{2}-70x-1136=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -70 for b, and -1136 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\left(-1\right)\left(-1136\right)}}{2\left(-1\right)}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\left(-1\right)\left(-1136\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-70\right)±\sqrt{4900+4\left(-1136\right)}}{2\left(-1\right)}$$

Multiply 4 times -1136.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4544}}{2\left(-1\right)}$$

Add 4900 to -4544.

$$x=\frac{-\left(-70\right)±\sqrt{356}}{2\left(-1\right)}$$

Take the square root of 356.

$$x=\frac{-\left(-70\right)±2\sqrt{89}}{2\left(-1\right)}$$

The opposite of -70 is 70.

$$x=\frac{70±2\sqrt{89}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{70±2\sqrt{89}}{-2}$$

Now solve the equation x=\frac{70±2\sqrt{89}}{-2} when ± is plus. Add 70 to 2\sqrt{89}.

$$x=\frac{2\sqrt{89}+70}{-2}$$

Divide 70+2\sqrt{89} by -2.

$$x=-\left(\sqrt{89}+35\right)$$

Now solve the equation x=\frac{70±2\sqrt{89}}{-2} when ± is minus. Subtract 2\sqrt{89} from 70.

$$x=\frac{70-2\sqrt{89}}{-2}$$

Divide 70-2\sqrt{89} by -2.

$$x=\sqrt{89}-35$$

The equation is now solved.

$$x=-\left(\sqrt{89}+35\right) x=\sqrt{89}-35$$

Steps for Completing the Square

Multiply 16 and 121 to get 1936.

$$1936=\left(80+x\right)\left(10-x\right)$$

Use the distributive property to multiply 80+x by 10-x and combine like terms.

$$1936=800-70x-x^{2}$$

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

$$800-70x-x^{2}=1936$$

Subtract 800 from both sides.

$$-70x-x^{2}=1936-800$$

Subtract 800 from 1936 to get 1136.

$$-70x-x^{2}=1136$$

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.

$$-x^{2}-70x=1136$$

Divide both sides by -1.

$$\frac{-x^{2}-70x}{-1}=\frac{1136}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{70}{-1}\right)x=\frac{1136}{-1}$$

Divide -70 by -1.

$$x^{2}+70x=\frac{1136}{-1}$$

Divide 1136 by -1.

$$x^{2}+70x=-1136$$

Divide 70, the coefficient of the x term, by 2 to get 35. Then add the square of 35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+70x+35^{2}=-1136+35^{2}$$

Square 35.

$$x^{2}+70x+1225=-1136+1225$$

Add -1136 to 1225.

$$x^{2}+70x+1225=89$$

Factor x^{2}+70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+35\right)^{2}=89$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+35\right)^{2}}=\sqrt{89}$$

Simplify.

$$x+35=\sqrt{89} x+35=-\sqrt{89}$$

Subtract 35 from both sides of the equation.

$$x=\sqrt{89}-35 x=-\sqrt{89}-35$$