Solve 100=75x+25/2x^2

$$100=75x+ \frac{ 25 }{ 2 } { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\sqrt{17}-3\approx 1.123105626$$

$$x=-\left(\sqrt{17}+3\right)\approx -7.123105626$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$75x+\frac{25}{2}x^{2}=100$$

Subtract 100 from both sides.

$$75x+\frac{25}{2}x^{2}-100=0$$

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.

$$\frac{25}{2}x^{2}+75x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{25}{2} for a, 75 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-75±\sqrt{75^{2}-4\times \frac{25}{2}\left(-100\right)}}{2\times \frac{25}{2}}$$

Square 75.

$$x=\frac{-75±\sqrt{5625-4\times \frac{25}{2}\left(-100\right)}}{2\times \frac{25}{2}}$$

Multiply -4 times \frac{25}{2}.

$$x=\frac{-75±\sqrt{5625-50\left(-100\right)}}{2\times \frac{25}{2}}$$

Multiply -50 times -100.

$$x=\frac{-75±\sqrt{5625+5000}}{2\times \frac{25}{2}}$$

Add 5625 to 5000.

$$x=\frac{-75±\sqrt{10625}}{2\times \frac{25}{2}}$$

Take the square root of 10625.

$$x=\frac{-75±25\sqrt{17}}{2\times \frac{25}{2}}$$

Multiply 2 times \frac{25}{2}.

$$x=\frac{-75±25\sqrt{17}}{25}$$

Now solve the equation x=\frac{-75±25\sqrt{17}}{25} when ± is plus. Add -75 to 25\sqrt{17}.

$$x=\frac{25\sqrt{17}-75}{25}$$

Divide -75+25\sqrt{17} by 25.

$$x=\sqrt{17}-3$$

Now solve the equation x=\frac{-75±25\sqrt{17}}{25} when ± is minus. Subtract 25\sqrt{17} from -75.

$$x=\frac{-25\sqrt{17}-75}{25}$$

Divide -75-25\sqrt{17} by 25.

$$x=-\sqrt{17}-3$$

The equation is now solved.

$$x=\sqrt{17}-3 x=-\sqrt{17}-3$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$75x+\frac{25}{2}x^{2}=100$$

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.

$$\frac{25}{2}x^{2}+75x=100$$

Divide both sides of the equation by \frac{25}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{25}{2}x^{2}+75x}{\frac{25}{2}}=\frac{100}{\frac{25}{2}}$$

Dividing by \frac{25}{2} undoes the multiplication by \frac{25}{2}.

$$x^{2}+\frac{75}{\frac{25}{2}}x=\frac{100}{\frac{25}{2}}$$

Divide 75 by \frac{25}{2} by multiplying 75 by the reciprocal of \frac{25}{2}.

$$x^{2}+6x=\frac{100}{\frac{25}{2}}$$

Divide 100 by \frac{25}{2} by multiplying 100 by the reciprocal of \frac{25}{2}.

$$x^{2}+6x=8$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=8+3^{2}$$

Square 3.

$$x^{2}+6x+9=8+9$$

Add 8 to 9.

$$x^{2}+6x+9=17$$

Factor x^{2}+6x+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}.

$$\left(x+3\right)^{2}=17$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{17}$$

Simplify.

$$x+3=\sqrt{17} x+3=-\sqrt{17}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{17}-3 x=-\sqrt{17}-3$$

Solve for x

$$x=\sqrt{17}-3\approx 1.123105626$$

$$x=-\sqrt{17}-3\approx -7.123105626$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$75x+\frac{25}{2}x^{2}=100$$

Subtract 100 from both sides.

$$75x+\frac{25}{2}x^{2}-100=0$$

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.

$$\frac{25}{2}x^{2}+75x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{25}{2} for a, 75 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-75±\sqrt{75^{2}-4\times \frac{25}{2}\left(-100\right)}}{2\times \frac{25}{2}}$$

Square 75.

$$x=\frac{-75±\sqrt{5625-4\times \frac{25}{2}\left(-100\right)}}{2\times \frac{25}{2}}$$

Multiply -4 times \frac{25}{2}.

$$x=\frac{-75±\sqrt{5625-50\left(-100\right)}}{2\times \frac{25}{2}}$$

Multiply -50 times -100.

$$x=\frac{-75±\sqrt{5625+5000}}{2\times \frac{25}{2}}$$

Add 5625 to 5000.

$$x=\frac{-75±\sqrt{10625}}{2\times \frac{25}{2}}$$

Take the square root of 10625.

$$x=\frac{-75±25\sqrt{17}}{2\times \frac{25}{2}}$$

Multiply 2 times \frac{25}{2}.

$$x=\frac{-75±25\sqrt{17}}{25}$$

Now solve the equation x=\frac{-75±25\sqrt{17}}{25} when ± is plus. Add -75 to 25\sqrt{17}.

$$x=\frac{25\sqrt{17}-75}{25}$$

Divide -75+25\sqrt{17} by 25.

$$x=\sqrt{17}-3$$

Now solve the equation x=\frac{-75±25\sqrt{17}}{25} when ± is minus. Subtract 25\sqrt{17} from -75.

$$x=\frac{-25\sqrt{17}-75}{25}$$

Divide -75-25\sqrt{17} by 25.

$$x=-\sqrt{17}-3$$

The equation is now solved.

$$x=\sqrt{17}-3 x=-\sqrt{17}-3$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$75x+\frac{25}{2}x^{2}=100$$

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.

$$\frac{25}{2}x^{2}+75x=100$$

Divide both sides of the equation by \frac{25}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{25}{2}x^{2}+75x}{\frac{25}{2}}=\frac{100}{\frac{25}{2}}$$

Dividing by \frac{25}{2} undoes the multiplication by \frac{25}{2}.

$$x^{2}+\frac{75}{\frac{25}{2}}x=\frac{100}{\frac{25}{2}}$$

Divide 75 by \frac{25}{2} by multiplying 75 by the reciprocal of \frac{25}{2}.

$$x^{2}+6x=\frac{100}{\frac{25}{2}}$$

Divide 100 by \frac{25}{2} by multiplying 100 by the reciprocal of \frac{25}{2}.

$$x^{2}+6x=8$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=8+3^{2}$$

Square 3.

$$x^{2}+6x+9=8+9$$

Add 8 to 9.

$$x^{2}+6x+9=17$$

Factor x^{2}+6x+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}.

$$\left(x+3\right)^{2}=17$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{17}$$

Simplify.

$$x+3=\sqrt{17} x+3=-\sqrt{17}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{17}-3 x=-\sqrt{17}-3$$