Solve 0.75x^2+15x-727.48=0

$$0.75 { x }^{ 2 } +15x-727.48=0$$

Solve for x

$$x=\frac{2\sqrt{60186}}{15}-10\approx 22.710446853$$

$$x=-\frac{2\sqrt{60186}}{15}-10\approx -42.710446853$$

Steps Using the Quadratic Formula

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.

$$0.75x^{2}+15x-727.48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.75 for a, 15 for b, and -727.48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 0.75\left(-727.48\right)}}{2\times 0.75}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 0.75\left(-727.48\right)}}{2\times 0.75}$$

Multiply -4 times 0.75.

$$x=\frac{-15±\sqrt{225-3\left(-727.48\right)}}{2\times 0.75}$$

Multiply -3 times -727.48.

$$x=\frac{-15±\sqrt{225+2182.44}}{2\times 0.75}$$

Add 225 to 2182.44.

$$x=\frac{-15±\sqrt{2407.44}}{2\times 0.75}$$

Take the square root of 2407.44.

$$x=\frac{-15±\frac{\sqrt{60186}}{5}}{2\times 0.75}$$

Multiply 2 times 0.75.

$$x=\frac{-15±\frac{\sqrt{60186}}{5}}{1.5}$$

Now solve the equation x=\frac{-15±\frac{\sqrt{60186}}{5}}{1.5} when ± is plus. Add -15 to \frac{\sqrt{60186}}{5}.

$$x=\frac{\frac{\sqrt{60186}}{5}-15}{1.5}$$

Divide -15+\frac{\sqrt{60186}}{5} by 1.5 by multiplying -15+\frac{\sqrt{60186}}{5} by the reciprocal of 1.5.

$$x=\frac{2\sqrt{60186}}{15}-10$$

Now solve the equation x=\frac{-15±\frac{\sqrt{60186}}{5}}{1.5} when ± is minus. Subtract \frac{\sqrt{60186}}{5} from -15.

$$x=\frac{-\frac{\sqrt{60186}}{5}-15}{1.5}$$

Divide -15-\frac{\sqrt{60186}}{5} by 1.5 by multiplying -15-\frac{\sqrt{60186}}{5} by the reciprocal of 1.5.

$$x=-\frac{2\sqrt{60186}}{15}-10$$

The equation is now solved.

$$x=\frac{2\sqrt{60186}}{15}-10 x=-\frac{2\sqrt{60186}}{15}-10$$

Steps for Completing the Square

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.

$$0.75x^{2}+15x-727.48=0$$

Add 727.48 to both sides of the equation.

$$0.75x^{2}+15x-727.48-\left(-727.48\right)=-\left(-727.48\right)$$

Subtracting -727.48 from itself leaves 0.

$$0.75x^{2}+15x=-\left(-727.48\right)$$

Subtract -727.48 from 0.

$$0.75x^{2}+15x=727.48$$

Divide both sides of the equation by 0.75, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.75x^{2}+15x}{0.75}=\frac{727.48}{0.75}$$

Dividing by 0.75 undoes the multiplication by 0.75.

$$x^{2}+\frac{15}{0.75}x=\frac{727.48}{0.75}$$

Divide 15 by 0.75 by multiplying 15 by the reciprocal of 0.75.

$$x^{2}+20x=\frac{727.48}{0.75}$$

Divide 727.48 by 0.75 by multiplying 727.48 by the reciprocal of 0.75.

$$x^{2}+20x=\frac{72748}{75}$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=\frac{72748}{75}+10^{2}$$

Square 10.

$$x^{2}+20x+100=\frac{72748}{75}+100$$

Add \frac{72748}{75} to 100.

$$x^{2}+20x+100=\frac{80248}{75}$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=\frac{80248}{75}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{\frac{80248}{75}}$$

Simplify.

$$x+10=\frac{2\sqrt{60186}}{15} x+10=-\frac{2\sqrt{60186}}{15}$$

Subtract 10 from both sides of the equation.

$$x=\frac{2\sqrt{60186}}{15}-10 x=-\frac{2\sqrt{60186}}{15}-10$$