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

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

Solve for x (complex solution)

$$x=\frac{9+\sqrt{39919}i}{10}\approx 0.9+19.979739738i$$

$$x=\frac{-\sqrt{39919}i+9}{10}\approx 0.9-19.979739738i$$

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}.

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

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

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

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

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

Multiply 10 times -1000.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-10000}}{2\left(-\frac{5}{2}\right)}$$

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

$$x=\frac{-\frac{9}{2}±\sqrt{-\frac{39919}{4}}}{2\left(-\frac{5}{2}\right)}$$

Take the square root of -\frac{39919}{4}.

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{2\left(-\frac{5}{2}\right)}$$

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

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5}$$

Now solve the equation x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5} when ± is plus. Add -\frac{9}{2} to \frac{i\sqrt{39919}}{2}.

$$x=\frac{-9+\sqrt{39919}i}{-5\times 2}$$

Divide \frac{-9+i\sqrt{39919}}{2} by -5.

$$x=\frac{-\sqrt{39919}i+9}{10}$$

Now solve the equation x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5} when ± is minus. Subtract \frac{i\sqrt{39919}}{2} from -\frac{9}{2}.

$$x=\frac{-\sqrt{39919}i-9}{-5\times 2}$$

Divide \frac{-9-i\sqrt{39919}}{2} by -5.

$$x=\frac{9+\sqrt{39919}i}{10}$$

The equation is now solved.

$$x=\frac{-\sqrt{39919}i+9}{10} x=\frac{9+\sqrt{39919}i}{10}$$

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}.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=-\frac{39919}{100}$$

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}.

$$\left(x-\frac{9}{10}\right)^{2}=-\frac{39919}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{10}\right)^{2}}=\sqrt{-\frac{39919}{100}}$$

Simplify.

$$x-\frac{9}{10}=\frac{\sqrt{39919}i}{10} x-\frac{9}{10}=-\frac{\sqrt{39919}i}{10}$$

Add \frac{9}{10} to both sides of the equation.

$$x=\frac{9+\sqrt{39919}i}{10} x=\frac{-\sqrt{39919}i+9}{10}$$