Solve (x(1.25x+15)-50*40)*30+x(1.25x+15)*100=6420000

$$(x(1.25x+15)-50 \times 40) \times 30+x(1.25x+15) \times 100=6420000$$

Solve for x

$$x=\frac{6\sqrt{187369}}{13}-6\approx 193.782189088$$

$$x=-\frac{6\sqrt{187369}}{13}-6\approx -205.782189088$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 1.25x+15.

$$\left(1.25x^{2}+15x-50\times 40\right)\times 30+x\left(1.25x+15\right)\times 100=6420000$$

Multiply 50 and 40 to get 2000.

$$\left(1.25x^{2}+15x-2000\right)\times 30+x\left(1.25x+15\right)\times 100=6420000$$

Use the distributive property to multiply 1.25x^{2}+15x-2000 by 30.

$$37.5x^{2}+450x-60000+x\left(1.25x+15\right)\times 100=6420000$$

Use the distributive property to multiply x by 1.25x+15.

$$37.5x^{2}+450x-60000+\left(1.25x^{2}+15x\right)\times 100=6420000$$

Use the distributive property to multiply 1.25x^{2}+15x by 100.

$$37.5x^{2}+450x-60000+125x^{2}+1500x=6420000$$

Combine 37.5x^{2} and 125x^{2} to get 162.5x^{2}.

$$162.5x^{2}+450x-60000+1500x=6420000$$

Combine 450x and 1500x to get 1950x.

$$162.5x^{2}+1950x-60000=6420000$$

Subtract 6420000 from both sides.

$$162.5x^{2}+1950x-60000-6420000=0$$

Subtract 6420000 from -60000 to get -6480000.

$$162.5x^{2}+1950x-6480000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 162.5 for a, 1950 for b, and -6480000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1950±\sqrt{1950^{2}-4\times 162.5\left(-6480000\right)}}{2\times 162.5}$$

Square 1950.

$$x=\frac{-1950±\sqrt{3802500-4\times 162.5\left(-6480000\right)}}{2\times 162.5}$$

Multiply -4 times 162.5.

$$x=\frac{-1950±\sqrt{3802500-650\left(-6480000\right)}}{2\times 162.5}$$

Multiply -650 times -6480000.

$$x=\frac{-1950±\sqrt{3802500+4212000000}}{2\times 162.5}$$

Add 3802500 to 4212000000.

$$x=\frac{-1950±\sqrt{4215802500}}{2\times 162.5}$$

Take the square root of 4215802500.

$$x=\frac{-1950±150\sqrt{187369}}{2\times 162.5}$$

Multiply 2 times 162.5.

$$x=\frac{-1950±150\sqrt{187369}}{325}$$

Now solve the equation x=\frac{-1950±150\sqrt{187369}}{325} when ± is plus. Add -1950 to 150\sqrt{187369}.

$$x=\frac{150\sqrt{187369}-1950}{325}$$

Divide -1950+150\sqrt{187369} by 325.

$$x=\frac{6\sqrt{187369}}{13}-6$$

Now solve the equation x=\frac{-1950±150\sqrt{187369}}{325} when ± is minus. Subtract 150\sqrt{187369} from -1950.

$$x=\frac{-150\sqrt{187369}-1950}{325}$$

Divide -1950-150\sqrt{187369} by 325.

$$x=-\frac{6\sqrt{187369}}{13}-6$$

The equation is now solved.

$$x=\frac{6\sqrt{187369}}{13}-6 x=-\frac{6\sqrt{187369}}{13}-6$$

Steps for Completing the Square

Use the distributive property to multiply x by 1.25x+15.

$$\left(1.25x^{2}+15x-50\times 40\right)\times 30+x\left(1.25x+15\right)\times 100=6420000$$

Multiply 50 and 40 to get 2000.

$$\left(1.25x^{2}+15x-2000\right)\times 30+x\left(1.25x+15\right)\times 100=6420000$$

Use the distributive property to multiply 1.25x^{2}+15x-2000 by 30.

$$37.5x^{2}+450x-60000+x\left(1.25x+15\right)\times 100=6420000$$

Use the distributive property to multiply x by 1.25x+15.

$$37.5x^{2}+450x-60000+\left(1.25x^{2}+15x\right)\times 100=6420000$$

Use the distributive property to multiply 1.25x^{2}+15x by 100.

$$37.5x^{2}+450x-60000+125x^{2}+1500x=6420000$$

Combine 37.5x^{2} and 125x^{2} to get 162.5x^{2}.

$$162.5x^{2}+450x-60000+1500x=6420000$$

Combine 450x and 1500x to get 1950x.

$$162.5x^{2}+1950x-60000=6420000$$

Add 60000 to both sides.

$$162.5x^{2}+1950x=6420000+60000$$

Add 6420000 and 60000 to get 6480000.

$$162.5x^{2}+1950x=6480000$$

Divide both sides of the equation by 162.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{162.5x^{2}+1950x}{162.5}=\frac{6480000}{162.5}$$

Dividing by 162.5 undoes the multiplication by 162.5.

$$x^{2}+\frac{1950}{162.5}x=\frac{6480000}{162.5}$$

Divide 1950 by 162.5 by multiplying 1950 by the reciprocal of 162.5.

$$x^{2}+12x=\frac{6480000}{162.5}$$

Divide 6480000 by 162.5 by multiplying 6480000 by the reciprocal of 162.5.

$$x^{2}+12x=\frac{518400}{13}$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=\frac{518400}{13}+6^{2}$$

Square 6.

$$x^{2}+12x+36=\frac{518400}{13}+36$$

Add \frac{518400}{13} to 36.

$$x^{2}+12x+36=\frac{518868}{13}$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=\frac{518868}{13}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{\frac{518868}{13}}$$

Simplify.

$$x+6=\frac{6\sqrt{187369}}{13} x+6=-\frac{6\sqrt{187369}}{13}$$

Subtract 6 from both sides of the equation.

$$x=\frac{6\sqrt{187369}}{13}-6 x=-\frac{6\sqrt{187369}}{13}-6$$