Solve 40x(x+1)+20left(x+1right)^2+15x(x+1)+left(x+1right)^2*15*0.5=7260

$$40x(x+1)+20 { \left(x+1 \right) }^{ 2 } +15x(x+1)+ { \left(x+1 \right) }^{ 2 } \times 15 \times 0.5=7260$$

Solve for x

$$x = \frac{\sqrt{793} - 2}{3} \approx 8.720085227$$

$$x=\frac{-\sqrt{793}-2}{3}\approx -10.05341856$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40x by x+1.

$$40x^{2}+40x+20\left(x+1\right)^{2}+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$40x^{2}+40x+20\left(x^{2}+2x+1\right)+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Use the distributive property to multiply 20 by x^{2}+2x+1.

$$40x^{2}+40x+20x^{2}+40x+20+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Combine 40x^{2} and 20x^{2} to get 60x^{2}.

$$60x^{2}+40x+40x+20+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Combine 40x and 40x to get 80x.

$$60x^{2}+80x+20+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Use the distributive property to multiply 15x by x+1.

$$60x^{2}+80x+20+15x^{2}+15x+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Combine 60x^{2} and 15x^{2} to get 75x^{2}.

$$75x^{2}+80x+20+15x+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Combine 80x and 15x to get 95x.

$$75x^{2}+95x+20+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$75x^{2}+95x+20+\left(x^{2}+2x+1\right)\times 15\times 0.5=7260$$

Multiply 15 and 0.5 to get 7.5.

$$75x^{2}+95x+20+\left(x^{2}+2x+1\right)\times 7.5=7260$$

Use the distributive property to multiply x^{2}+2x+1 by 7.5.

$$75x^{2}+95x+20+7.5x^{2}+15x+7.5=7260$$

Combine 75x^{2} and 7.5x^{2} to get 82.5x^{2}.

$$82.5x^{2}+95x+20+15x+7.5=7260$$

Combine 95x and 15x to get 110x.

$$82.5x^{2}+110x+20+7.5=7260$$

Add 20 and 7.5 to get 27.5.

$$82.5x^{2}+110x+27.5=7260$$

Subtract 7260 from both sides.

$$82.5x^{2}+110x+27.5-7260=0$$

Subtract 7260 from 27.5 to get -7232.5.

$$82.5x^{2}+110x-7232.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 82.5 for a, 110 for b, and -7232.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-110±\sqrt{110^{2}-4\times 82.5\left(-7232.5\right)}}{2\times 82.5}$$

Square 110.

$$x=\frac{-110±\sqrt{12100-4\times 82.5\left(-7232.5\right)}}{2\times 82.5}$$

Multiply -4 times 82.5.

$$x=\frac{-110±\sqrt{12100-330\left(-7232.5\right)}}{2\times 82.5}$$

Multiply -330 times -7232.5.

$$x=\frac{-110±\sqrt{12100+2386725}}{2\times 82.5}$$

Add 12100 to 2386725.

$$x=\frac{-110±\sqrt{2398825}}{2\times 82.5}$$

Take the square root of 2398825.

$$x=\frac{-110±55\sqrt{793}}{2\times 82.5}$$

Multiply 2 times 82.5.

$$x=\frac{-110±55\sqrt{793}}{165}$$

Now solve the equation x=\frac{-110±55\sqrt{793}}{165} when ± is plus. Add -110 to 55\sqrt{793}.

$$x=\frac{55\sqrt{793}-110}{165}$$

Divide -110+55\sqrt{793} by 165.

$$x=\frac{\sqrt{793}-2}{3}$$

Now solve the equation x=\frac{-110±55\sqrt{793}}{165} when ± is minus. Subtract 55\sqrt{793} from -110.

$$x=\frac{-55\sqrt{793}-110}{165}$$

Divide -110-55\sqrt{793} by 165.

$$x=\frac{-\sqrt{793}-2}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{793}-2}{3} x=\frac{-\sqrt{793}-2}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 40x by x+1.

$$40x^{2}+40x+20\left(x+1\right)^{2}+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$40x^{2}+40x+20\left(x^{2}+2x+1\right)+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Use the distributive property to multiply 20 by x^{2}+2x+1.

$$40x^{2}+40x+20x^{2}+40x+20+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Combine 40x^{2} and 20x^{2} to get 60x^{2}.

$$60x^{2}+40x+40x+20+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Combine 40x and 40x to get 80x.

$$60x^{2}+80x+20+15x\left(x+1\right)+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Use the distributive property to multiply 15x by x+1.

$$60x^{2}+80x+20+15x^{2}+15x+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Combine 60x^{2} and 15x^{2} to get 75x^{2}.

$$75x^{2}+80x+20+15x+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Combine 80x and 15x to get 95x.

$$75x^{2}+95x+20+\left(x+1\right)^{2}\times 15\times 0.5=7260$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$75x^{2}+95x+20+\left(x^{2}+2x+1\right)\times 15\times 0.5=7260$$

Multiply 15 and 0.5 to get 7.5.

$$75x^{2}+95x+20+\left(x^{2}+2x+1\right)\times 7.5=7260$$

Use the distributive property to multiply x^{2}+2x+1 by 7.5.

$$75x^{2}+95x+20+7.5x^{2}+15x+7.5=7260$$

Combine 75x^{2} and 7.5x^{2} to get 82.5x^{2}.

$$82.5x^{2}+95x+20+15x+7.5=7260$$

Combine 95x and 15x to get 110x.

$$82.5x^{2}+110x+20+7.5=7260$$

Add 20 and 7.5 to get 27.5.

$$82.5x^{2}+110x+27.5=7260$$

Subtract 27.5 from both sides.

$$82.5x^{2}+110x=7260-27.5$$

Subtract 27.5 from 7260 to get 7232.5.

$$82.5x^{2}+110x=7232.5$$

Divide both sides of the equation by 82.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{82.5x^{2}+110x}{82.5}=\frac{7232.5}{82.5}$$

Dividing by 82.5 undoes the multiplication by 82.5.

$$x^{2}+\frac{110}{82.5}x=\frac{7232.5}{82.5}$$

Divide 110 by 82.5 by multiplying 110 by the reciprocal of 82.5.

$$x^{2}+\frac{4}{3}x=\frac{7232.5}{82.5}$$

Divide 7232.5 by 82.5 by multiplying 7232.5 by the reciprocal of 82.5.

$$x^{2}+\frac{4}{3}x=\frac{263}{3}$$

Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{3}x+\frac{2}{3}^{2}=\frac{263}{3}+\frac{2}{3}^{2}$$

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{263}{3}+\frac{4}{9}$$

Add \frac{263}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{793}{9}$$

Factor x^{2}+\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{3}\right)^{2}=\frac{793}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{\frac{793}{9}}$$

Simplify.

$$x+\frac{2}{3}=\frac{\sqrt{793}}{3} x+\frac{2}{3}=-\frac{\sqrt{793}}{3}$$

Subtract \frac{2}{3} from both sides of the equation.

$$x=\frac{\sqrt{793}-2}{3} x=\frac{-\sqrt{793}-2}{3}$$