Solve (x-3.5)*(x+38-2.5)=2535.75

$$(x-3.5) \times (x+38-2.5)=2535.75$$

Solve for x

$$x=-70$$

$$x=38$$

Steps Using the Quadratic Formula

Subtract 2.5 from 38 to get 35.5.

$$\left(x-3.5\right)\left(x+35.5\right)=2535.75$$

Use the distributive property to multiply x-3.5 by x+35.5 and combine like terms.

$$x^{2}+32x-124.25=2535.75$$

Subtract 2535.75 from both sides.

$$x^{2}+32x-124.25-2535.75=0$$

Subtract 2535.75 from -124.25 to get -2660.

$$x^{2}+32x-2660=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 32 for b, and -2660 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-2660\right)}}{2}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\left(-2660\right)}}{2}$$

Multiply -4 times -2660.

$$x=\frac{-32±\sqrt{1024+10640}}{2}$$

Add 1024 to 10640.

$$x=\frac{-32±\sqrt{11664}}{2}$$

Take the square root of 11664.

$$x=\frac{-32±108}{2}$$

Now solve the equation x=\frac{-32±108}{2} when ± is plus. Add -32 to 108.

$$x=\frac{76}{2}$$

Divide 76 by 2.

$$x=38$$

Now solve the equation x=\frac{-32±108}{2} when ± is minus. Subtract 108 from -32.

$$x=-\frac{140}{2}$$

Divide -140 by 2.

$$x=-70$$

The equation is now solved.

$$x=38 x=-70$$

Steps for Completing the Square

Subtract 2.5 from 38 to get 35.5.

$$\left(x-3.5\right)\left(x+35.5\right)=2535.75$$

Use the distributive property to multiply x-3.5 by x+35.5 and combine like terms.

$$x^{2}+32x-124.25=2535.75$$

Add 124.25 to both sides.

$$x^{2}+32x=2535.75+124.25$$

Add 2535.75 and 124.25 to get 2660.

$$x^{2}+32x=2660$$

Divide 32, the coefficient of the x term, by 2 to get 16. Then add the square of 16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+32x+16^{2}=2660+16^{2}$$

Square 16.

$$x^{2}+32x+256=2660+256$$

Add 2660 to 256.

$$x^{2}+32x+256=2916$$

Factor x^{2}+32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+16\right)^{2}=2916$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+16\right)^{2}}=\sqrt{2916}$$

Simplify.

$$x+16=54 x+16=-54$$

Subtract 16 from both sides of the equation.

$$x=38 x=-70$$