Solve 7x+5/2x^2-5/2x=1000

$$7x+ \frac{ 5 }{ 2 } { x }^{ 2 } - \frac{ 5 }{ 2 } x=1000$$

Solve for x

$$x = \frac{\sqrt{40081} - 9}{10} \approx 19.120239759$$

$$x=\frac{-\sqrt{40081}-9}{10}\approx -20.920239759$$

Steps Using the Quadratic Formula

Combine 7x and -\frac{5}{2}x to get \frac{9}{2}x.

$$\frac{9}{2}x+\frac{5}{2}x^{2}=1000$$

Subtract 1000 from both sides.

$$\frac{9}{2}x+\frac{5}{2}x^{2}-1000=0$$

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.

$$\frac{5}{2}x^{2}+\frac{9}{2}x-1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{5}{2} for a, \frac{9}{2} for b, and -1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{2}±\sqrt{\left(\frac{9}{2}\right)^{2}-4\times \frac{5}{2}\left(-1000\right)}}{2\times \frac{5}{2}}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-4\times \frac{5}{2}\left(-1000\right)}}{2\times \frac{5}{2}}$$

Multiply -4 times \frac{5}{2}.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-10\left(-1000\right)}}{2\times \frac{5}{2}}$$

Multiply -10 times -1000.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}+10000}}{2\times \frac{5}{2}}$$

Add \frac{81}{4} to 10000.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{40081}{4}}}{2\times \frac{5}{2}}$$

Take the square root of \frac{40081}{4}.

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{40081}}{2}}{2\times \frac{5}{2}}$$

Multiply 2 times \frac{5}{2}.

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{40081}}{2}}{5}$$

Now solve the equation x=\frac{-\frac{9}{2}±\frac{\sqrt{40081}}{2}}{5} when ± is plus. Add -\frac{9}{2} to \frac{\sqrt{40081}}{2}.

$$x=\frac{\sqrt{40081}-9}{2\times 5}$$

Divide \frac{-9+\sqrt{40081}}{2} by 5.

$$x=\frac{\sqrt{40081}-9}{10}$$

Now solve the equation x=\frac{-\frac{9}{2}±\frac{\sqrt{40081}}{2}}{5} when ± is minus. Subtract \frac{\sqrt{40081}}{2} from -\frac{9}{2}.

$$x=\frac{-\sqrt{40081}-9}{2\times 5}$$

Divide \frac{-9-\sqrt{40081}}{2} by 5.

$$x=\frac{-\sqrt{40081}-9}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{40081}-9}{10} x=\frac{-\sqrt{40081}-9}{10}$$

Steps for Completing the Square

Combine 7x and -\frac{5}{2}x to get \frac{9}{2}x.

$$\frac{9}{2}x+\frac{5}{2}x^{2}=1000$$

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.

$$\frac{5}{2}x^{2}+\frac{9}{2}x=1000$$

Divide both sides of the equation by \frac{5}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{5}{2}x^{2}+\frac{9}{2}x}{\frac{5}{2}}=\frac{1000}{\frac{5}{2}}$$

Dividing by \frac{5}{2} undoes the multiplication by \frac{5}{2}.

$$x^{2}+\frac{\frac{9}{2}}{\frac{5}{2}}x=\frac{1000}{\frac{5}{2}}$$

Divide \frac{9}{2} by \frac{5}{2} by multiplying \frac{9}{2} by the reciprocal of \frac{5}{2}.

$$x^{2}+\frac{9}{5}x=\frac{1000}{\frac{5}{2}}$$

Divide 1000 by \frac{5}{2} by multiplying 1000 by the reciprocal of \frac{5}{2}.

$$x^{2}+\frac{9}{5}x=400$$

Divide \frac{9}{5}, the coefficient of the x term, by 2 to get \frac{9}{10}. Then add the square of \frac{9}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{5}x+\left(\frac{9}{10}\right)^{2}=400+\left(\frac{9}{10}\right)^{2}$$

Square \frac{9}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{5}x+\frac{81}{100}=400+\frac{81}{100}$$

Add 400 to \frac{81}{100}.

$$x^{2}+\frac{9}{5}x+\frac{81}{100}=\frac{40081}{100}$$

Factor x^{2}+\frac{9}{5}x+\frac{81}{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}.

$$\left(x+\frac{9}{10}\right)^{2}=\frac{40081}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{10}\right)^{2}}=\sqrt{\frac{40081}{100}}$$

Simplify.

$$x+\frac{9}{10}=\frac{\sqrt{40081}}{10} x+\frac{9}{10}=-\frac{\sqrt{40081}}{10}$$

Subtract \frac{9}{10} from both sides of the equation.

$$x=\frac{\sqrt{40081}-9}{10} x=\frac{-\sqrt{40081}-9}{10}$$