Solve 100=30x+1/2*2.525*x^2

$$100=30x+ \frac{ 1 }{ 2 } \times 2.525 \times { x }^{ 2 }$$

Solve for x

$$x = \frac{40 \sqrt{1405} - 1200}{101} \approx 2.963694902$$

$$x=\frac{-40\sqrt{1405}-1200}{101}\approx -26.72607114$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 2.525 to get \frac{101}{80}.

$$100=30x+\frac{101}{80}x^{2}$$

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

$$30x+\frac{101}{80}x^{2}=100$$

Subtract 100 from both sides.

$$30x+\frac{101}{80}x^{2}-100=0$$

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.

$$\frac{101}{80}x^{2}+30x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{101}{80} for a, 30 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times \frac{101}{80}\left(-100\right)}}{2\times \frac{101}{80}}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times \frac{101}{80}\left(-100\right)}}{2\times \frac{101}{80}}$$

Multiply -4 times \frac{101}{80}.

$$x=\frac{-30±\sqrt{900-\frac{101}{20}\left(-100\right)}}{2\times \frac{101}{80}}$$

Multiply -\frac{101}{20} times -100.

$$x=\frac{-30±\sqrt{900+505}}{2\times \frac{101}{80}}$$

Add 900 to 505.

$$x=\frac{-30±\sqrt{1405}}{2\times \frac{101}{80}}$$

Multiply 2 times \frac{101}{80}.

$$x=\frac{-30±\sqrt{1405}}{\frac{101}{40}}$$

Now solve the equation x=\frac{-30±\sqrt{1405}}{\frac{101}{40}} when ± is plus. Add -30 to \sqrt{1405}.

$$x=\frac{\sqrt{1405}-30}{\frac{101}{40}}$$

Divide -30+\sqrt{1405} by \frac{101}{40} by multiplying -30+\sqrt{1405} by the reciprocal of \frac{101}{40}.

$$x=\frac{40\sqrt{1405}-1200}{101}$$

Now solve the equation x=\frac{-30±\sqrt{1405}}{\frac{101}{40}} when ± is minus. Subtract \sqrt{1405} from -30.

$$x=\frac{-\sqrt{1405}-30}{\frac{101}{40}}$$

Divide -30-\sqrt{1405} by \frac{101}{40} by multiplying -30-\sqrt{1405} by the reciprocal of \frac{101}{40}.

$$x=\frac{-40\sqrt{1405}-1200}{101}$$

The equation is now solved.

$$x=\frac{40\sqrt{1405}-1200}{101} x=\frac{-40\sqrt{1405}-1200}{101}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 2.525 to get \frac{101}{80}.

$$100=30x+\frac{101}{80}x^{2}$$

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

$$30x+\frac{101}{80}x^{2}=100$$

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.

$$\frac{101}{80}x^{2}+30x=100$$

Divide both sides of the equation by \frac{101}{80}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{101}{80}x^{2}+30x}{\frac{101}{80}}=\frac{100}{\frac{101}{80}}$$

Dividing by \frac{101}{80} undoes the multiplication by \frac{101}{80}.

$$x^{2}+\frac{30}{\frac{101}{80}}x=\frac{100}{\frac{101}{80}}$$

Divide 30 by \frac{101}{80} by multiplying 30 by the reciprocal of \frac{101}{80}.

$$x^{2}+\frac{2400}{101}x=\frac{100}{\frac{101}{80}}$$

Divide 100 by \frac{101}{80} by multiplying 100 by the reciprocal of \frac{101}{80}.

$$x^{2}+\frac{2400}{101}x=\frac{8000}{101}$$

Divide \frac{2400}{101}, the coefficient of the x term, by 2 to get \frac{1200}{101}. Then add the square of \frac{1200}{101} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2400}{101}x+\left(\frac{1200}{101}\right)^{2}=\frac{8000}{101}+\left(\frac{1200}{101}\right)^{2}$$

Square \frac{1200}{101} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2400}{101}x+\frac{1440000}{10201}=\frac{8000}{101}+\frac{1440000}{10201}$$

Add \frac{8000}{101} to \frac{1440000}{10201} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2400}{101}x+\frac{1440000}{10201}=\frac{2248000}{10201}$$

Factor x^{2}+\frac{2400}{101}x+\frac{1440000}{10201}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1200}{101}\right)^{2}=\frac{2248000}{10201}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1200}{101}\right)^{2}}=\sqrt{\frac{2248000}{10201}}$$

Simplify.

$$x+\frac{1200}{101}=\frac{40\sqrt{1405}}{101} x+\frac{1200}{101}=-\frac{40\sqrt{1405}}{101}$$

Subtract \frac{1200}{101} from both sides of the equation.

$$x=\frac{40\sqrt{1405}-1200}{101} x=\frac{-40\sqrt{1405}-1200}{101}$$