Solve 81*25=left(25+xright)left(71-2xright)

$$81 \times 25 = \left( 25+x \right) \left( 71-2x \right)$$

Solve for x (complex solution)

$$x=\frac{21+\sqrt{1559}i}{4}\approx 5.25+9.871043511i$$

$$x=\frac{-\sqrt{1559}i+21}{4}\approx 5.25-9.871043511i$$

Steps Using the Quadratic Formula

Multiply 81 and 25 to get 2025.

$$2025=\left(25+x\right)\left(71-2x\right)$$

Use the distributive property to multiply 25+x by 71-2x and combine like terms.

$$2025=1775+21x-2x^{2}$$

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

$$1775+21x-2x^{2}=2025$$

Subtract 2025 from both sides.

$$1775+21x-2x^{2}-2025=0$$

Subtract 2025 from 1775 to get -250.

$$-250+21x-2x^{2}=0$$

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.

$$-2x^{2}+21x-250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 21 for b, and -250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\left(-2\right)\left(-250\right)}}{2\left(-2\right)}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\left(-2\right)\left(-250\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-21±\sqrt{441+8\left(-250\right)}}{2\left(-2\right)}$$

Multiply 8 times -250.

$$x=\frac{-21±\sqrt{441-2000}}{2\left(-2\right)}$$

Add 441 to -2000.

$$x=\frac{-21±\sqrt{-1559}}{2\left(-2\right)}$$

Take the square root of -1559.

$$x=\frac{-21±\sqrt{1559}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-21±\sqrt{1559}i}{-4}$$

Now solve the equation x=\frac{-21±\sqrt{1559}i}{-4} when ± is plus. Add -21 to i\sqrt{1559}.

$$x=\frac{-21+\sqrt{1559}i}{-4}$$

Divide -21+i\sqrt{1559} by -4.

$$x=\frac{-\sqrt{1559}i+21}{4}$$

Now solve the equation x=\frac{-21±\sqrt{1559}i}{-4} when ± is minus. Subtract i\sqrt{1559} from -21.

$$x=\frac{-\sqrt{1559}i-21}{-4}$$

Divide -21-i\sqrt{1559} by -4.

$$x=\frac{21+\sqrt{1559}i}{4}$$

The equation is now solved.

$$x=\frac{-\sqrt{1559}i+21}{4} x=\frac{21+\sqrt{1559}i}{4}$$

Steps for Completing the Square

Multiply 81 and 25 to get 2025.

$$2025=\left(25+x\right)\left(71-2x\right)$$

Use the distributive property to multiply 25+x by 71-2x and combine like terms.

$$2025=1775+21x-2x^{2}$$

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

$$1775+21x-2x^{2}=2025$$

Subtract 1775 from both sides.

$$21x-2x^{2}=2025-1775$$

Subtract 1775 from 2025 to get 250.

$$21x-2x^{2}=250$$

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.

$$-2x^{2}+21x=250$$

Divide both sides by -2.

$$\frac{-2x^{2}+21x}{-2}=\frac{250}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{21}{-2}x=\frac{250}{-2}$$

Divide 21 by -2.

$$x^{2}-\frac{21}{2}x=\frac{250}{-2}$$

Divide 250 by -2.

$$x^{2}-\frac{21}{2}x=-125$$

Divide -\frac{21}{2}, the coefficient of the x term, by 2 to get -\frac{21}{4}. Then add the square of -\frac{21}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{21}{2}x+\left(-\frac{21}{4}\right)^{2}=-125+\left(-\frac{21}{4}\right)^{2}$$

Square -\frac{21}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{21}{2}x+\frac{441}{16}=-125+\frac{441}{16}$$

Add -125 to \frac{441}{16}.

$$x^{2}-\frac{21}{2}x+\frac{441}{16}=-\frac{1559}{16}$$

Factor x^{2}-\frac{21}{2}x+\frac{441}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{4}\right)^{2}=-\frac{1559}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{4}\right)^{2}}=\sqrt{-\frac{1559}{16}}$$

Simplify.

$$x-\frac{21}{4}=\frac{\sqrt{1559}i}{4} x-\frac{21}{4}=-\frac{\sqrt{1559}i}{4}$$

Add \frac{21}{4} to both sides of the equation.

$$x=\frac{21+\sqrt{1559}i}{4} x=\frac{-\sqrt{1559}i+21}{4}$$