Solve (19+x)(75+x)=75*27

$$( 19 + x ) ( 75 + x ) = 75 \times 27$$

Solve for x

$$x=-100$$

$$x=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 19+x by 75+x and combine like terms.

$$1425+94x+x^{2}=75\times 27$$

Multiply 75 and 27 to get 2025.

$$1425+94x+x^{2}=2025$$

Subtract 2025 from both sides.

$$1425+94x+x^{2}-2025=0$$

Subtract 2025 from 1425 to get -600.

$$-600+94x+x^{2}=0$$

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.

$$x^{2}+94x-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 94 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-94±\sqrt{94^{2}-4\left(-600\right)}}{2}$$

Square 94.

$$x=\frac{-94±\sqrt{8836-4\left(-600\right)}}{2}$$

Multiply -4 times -600.

$$x=\frac{-94±\sqrt{8836+2400}}{2}$$

Add 8836 to 2400.

$$x=\frac{-94±\sqrt{11236}}{2}$$

Take the square root of 11236.

$$x=\frac{-94±106}{2}$$

Now solve the equation x=\frac{-94±106}{2} when ± is plus. Add -94 to 106.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{-94±106}{2} when ± is minus. Subtract 106 from -94.

$$x=-\frac{200}{2}$$

Divide -200 by 2.

$$x=-100$$

The equation is now solved.

$$x=6 x=-100$$

Steps for Completing the Square

Use the distributive property to multiply 19+x by 75+x and combine like terms.

$$1425+94x+x^{2}=75\times 27$$

Multiply 75 and 27 to get 2025.

$$1425+94x+x^{2}=2025$$

Subtract 1425 from both sides.

$$94x+x^{2}=2025-1425$$

Subtract 1425 from 2025 to get 600.

$$94x+x^{2}=600$$

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.

$$x^{2}+94x=600$$

Divide 94, the coefficient of the x term, by 2 to get 47. Then add the square of 47 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+94x+47^{2}=600+47^{2}$$

Square 47.

$$x^{2}+94x+2209=600+2209$$

Add 600 to 2209.

$$x^{2}+94x+2209=2809$$

Factor x^{2}+94x+2209. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+47\right)^{2}=2809$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+47\right)^{2}}=\sqrt{2809}$$

Simplify.

$$x+47=53 x+47=-53$$

Subtract 47 from both sides of the equation.

$$x=6 x=-100$$