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

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

Solve for x

$$x = -\frac{131}{2} = -65\frac{1}{2} = -65.5$$

$$x=0$$

Steps Using the Quadratic Formula

Multiply 81 and 25 to get 2025.

$$2025=\left(81+2x\right)\left(25+x\right)$$

Use the distributive property to multiply 81+2x by 25+x and combine like terms.

$$2025=2025+131x+2x^{2}$$

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

$$2025+131x+2x^{2}=2025$$

Subtract 2025 from both sides.

$$2025+131x+2x^{2}-2025=0$$

Subtract 2025 from 2025 to get 0.

$$131x+2x^{2}=0$$

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.

$$2x^{2}+131x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 131 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-131±\sqrt{131^{2}}}{2\times 2}$$

Take the square root of 131^{2}.

$$x=\frac{-131±131}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-131±131}{4}$$

Now solve the equation x=\frac{-131±131}{4} when ± is plus. Add -131 to 131.

$$x=\frac{0}{4}$$

Divide 0 by 4.

$$x=0$$

Now solve the equation x=\frac{-131±131}{4} when ± is minus. Subtract 131 from -131.

$$x=-\frac{262}{4}$$

Reduce the fraction \frac{-262}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{131}{2}$$

The equation is now solved.

$$x=0 x=-\frac{131}{2}$$

Steps for Completing the Square

Multiply 81 and 25 to get 2025.

$$2025=\left(81+2x\right)\left(25+x\right)$$

Use the distributive property to multiply 81+2x by 25+x and combine like terms.

$$2025=2025+131x+2x^{2}$$

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

$$2025+131x+2x^{2}=2025$$

Subtract 2025 from both sides.

$$131x+2x^{2}=2025-2025$$

Subtract 2025 from 2025 to get 0.

$$131x+2x^{2}=0$$

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.

$$2x^{2}+131x=0$$

Divide both sides by 2.

$$\frac{2x^{2}+131x}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{131}{2}x=\frac{0}{2}$$

Divide 0 by 2.

$$x^{2}+\frac{131}{2}x=0$$

Divide \frac{131}{2}, the coefficient of the x term, by 2 to get \frac{131}{4}. Then add the square of \frac{131}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{131}{2}x+\left(\frac{131}{4}\right)^{2}=\left(\frac{131}{4}\right)^{2}$$

Square \frac{131}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{131}{2}x+\frac{17161}{16}=\frac{17161}{16}$$

Factor x^{2}+\frac{131}{2}x+\frac{17161}{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}.

$$\left(x+\frac{131}{4}\right)^{2}=\frac{17161}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{131}{4}\right)^{2}}=\sqrt{\frac{17161}{16}}$$

Simplify.

$$x+\frac{131}{4}=\frac{131}{4} x+\frac{131}{4}=-\frac{131}{4}$$

Subtract \frac{131}{4} from both sides of the equation.

$$x=0 x=-\frac{131}{2}$$