Solve 81*25=(71+2x)*(25+x)

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

Solve for x

$$x = -\frac{125}{2} = -62\frac{1}{2} = -62.5$$

$$x=2$$

Steps Using the Quadratic Formula

Multiply 81 and 25 to get 2025.

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

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

$$2025=1775+121x+2x^{2}$$

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

$$1775+121x+2x^{2}=2025$$

Subtract 2025 from both sides.

$$1775+121x+2x^{2}-2025=0$$

Subtract 2025 from 1775 to get -250.

$$-250+121x+2x^{2}=0$$

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.

$$2x^{2}+121x-250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 121 for b, and -250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-121±\sqrt{121^{2}-4\times 2\left(-250\right)}}{2\times 2}$$

Square 121.

$$x=\frac{-121±\sqrt{14641-4\times 2\left(-250\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-121±\sqrt{14641-8\left(-250\right)}}{2\times 2}$$

Multiply -8 times -250.

$$x=\frac{-121±\sqrt{14641+2000}}{2\times 2}$$

Add 14641 to 2000.

$$x=\frac{-121±\sqrt{16641}}{2\times 2}$$

Take the square root of 16641.

$$x=\frac{-121±129}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-121±129}{4}$$

Now solve the equation x=\frac{-121±129}{4} when ± is plus. Add -121 to 129.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{-121±129}{4} when ± is minus. Subtract 129 from -121.

$$x=-\frac{250}{4}$$

Reduce the fraction \frac{-250}{4} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

Steps for Completing the Square

Multiply 81 and 25 to get 2025.

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

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

$$2025=1775+121x+2x^{2}$$

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

$$1775+121x+2x^{2}=2025$$

Subtract 1775 from both sides.

$$121x+2x^{2}=2025-1775$$

Subtract 1775 from 2025 to get 250.

$$121x+2x^{2}=250$$

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.

$$2x^{2}+121x=250$$

Divide both sides by 2.

$$\frac{2x^{2}+121x}{2}=\frac{250}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{121}{2}x=\frac{250}{2}$$

Divide 250 by 2.

$$x^{2}+\frac{121}{2}x=125$$

Divide \frac{121}{2}, the coefficient of the x term, by 2 to get \frac{121}{4}. Then add the square of \frac{121}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{121}{2}x+\left(\frac{121}{4}\right)^{2}=125+\left(\frac{121}{4}\right)^{2}$$

Square \frac{121}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{121}{2}x+\frac{14641}{16}=125+\frac{14641}{16}$$

Add 125 to \frac{14641}{16}.

$$x^{2}+\frac{121}{2}x+\frac{14641}{16}=\frac{16641}{16}$$

Factor x^{2}+\frac{121}{2}x+\frac{14641}{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}.

$$\left(x+\frac{121}{4}\right)^{2}=\frac{16641}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{121}{4}\right)^{2}}=\sqrt{\frac{16641}{16}}$$

Simplify.

$$x+\frac{121}{4}=\frac{129}{4} x+\frac{121}{4}=-\frac{129}{4}$$

Subtract \frac{121}{4} from both sides of the equation.

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