Solve (x+250)*(5x-45)=1200

$$(x+250) \times (5x-45)=1200$$

Solve for x

$$x = \frac{\sqrt{68041} - 241}{2} \approx 9.923349137$$

$$x=\frac{-\sqrt{68041}-241}{2}\approx -250.923349137$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+250 by 5x-45 and combine like terms.

$$5x^{2}+1205x-11250=1200$$

Subtract 1200 from both sides.

$$5x^{2}+1205x-11250-1200=0$$

Subtract 1200 from -11250 to get -12450.

$$5x^{2}+1205x-12450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 1205 for b, and -12450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1205±\sqrt{1205^{2}-4\times 5\left(-12450\right)}}{2\times 5}$$

Square 1205.

$$x=\frac{-1205±\sqrt{1452025-4\times 5\left(-12450\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-1205±\sqrt{1452025-20\left(-12450\right)}}{2\times 5}$$

Multiply -20 times -12450.

$$x=\frac{-1205±\sqrt{1452025+249000}}{2\times 5}$$

Add 1452025 to 249000.

$$x=\frac{-1205±\sqrt{1701025}}{2\times 5}$$

Take the square root of 1701025.

$$x=\frac{-1205±5\sqrt{68041}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-1205±5\sqrt{68041}}{10}$$

Now solve the equation x=\frac{-1205±5\sqrt{68041}}{10} when ± is plus. Add -1205 to 5\sqrt{68041}.

$$x=\frac{5\sqrt{68041}-1205}{10}$$

Divide -1205+5\sqrt{68041} by 10.

$$x=\frac{\sqrt{68041}-241}{2}$$

Now solve the equation x=\frac{-1205±5\sqrt{68041}}{10} when ± is minus. Subtract 5\sqrt{68041} from -1205.

$$x=\frac{-5\sqrt{68041}-1205}{10}$$

Divide -1205-5\sqrt{68041} by 10.

$$x=\frac{-\sqrt{68041}-241}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{68041}-241}{2} x=\frac{-\sqrt{68041}-241}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x+250 by 5x-45 and combine like terms.

$$5x^{2}+1205x-11250=1200$$

Add 11250 to both sides.

$$5x^{2}+1205x=1200+11250$$

Add 1200 and 11250 to get 12450.

$$5x^{2}+1205x=12450$$

Divide both sides by 5.

$$\frac{5x^{2}+1205x}{5}=\frac{12450}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{1205}{5}x=\frac{12450}{5}$$

Divide 1205 by 5.

$$x^{2}+241x=\frac{12450}{5}$$

Divide 12450 by 5.

$$x^{2}+241x=2490$$

Divide 241, the coefficient of the x term, by 2 to get \frac{241}{2}. Then add the square of \frac{241}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+241x+\left(\frac{241}{2}\right)^{2}=2490+\left(\frac{241}{2}\right)^{2}$$

Square \frac{241}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+241x+\frac{58081}{4}=2490+\frac{58081}{4}$$

Add 2490 to \frac{58081}{4}.

$$x^{2}+241x+\frac{58081}{4}=\frac{68041}{4}$$

Factor x^{2}+241x+\frac{58081}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{241}{2}\right)^{2}=\frac{68041}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{241}{2}\right)^{2}}=\sqrt{\frac{68041}{4}}$$

Simplify.

$$x+\frac{241}{2}=\frac{\sqrt{68041}}{2} x+\frac{241}{2}=-\frac{\sqrt{68041}}{2}$$

Subtract \frac{241}{2} from both sides of the equation.

$$x=\frac{\sqrt{68041}-241}{2} x=\frac{-\sqrt{68041}-241}{2}$$