Solve 7x^2+9268x-61800=0

$$7 { x }^{ 2 } +9268x-61800=0$$

Solve for x

$$x=\frac{2\sqrt{5476639}}{7}-662\approx 6.634856576$$

$$x=-\frac{2\sqrt{5476639}}{7}-662\approx -1330.634856576$$

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.

$$7x^{2}+9268x-61800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 9268 for b, and -61800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9268±\sqrt{9268^{2}-4\times 7\left(-61800\right)}}{2\times 7}$$

Square 9268.

$$x=\frac{-9268±\sqrt{85895824-4\times 7\left(-61800\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-9268±\sqrt{85895824-28\left(-61800\right)}}{2\times 7}$$

Multiply -28 times -61800.

$$x=\frac{-9268±\sqrt{85895824+1730400}}{2\times 7}$$

Add 85895824 to 1730400.

$$x=\frac{-9268±\sqrt{87626224}}{2\times 7}$$

Take the square root of 87626224.

$$x=\frac{-9268±4\sqrt{5476639}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-9268±4\sqrt{5476639}}{14}$$

Now solve the equation x=\frac{-9268±4\sqrt{5476639}}{14} when ± is plus. Add -9268 to 4\sqrt{5476639}.

$$x=\frac{4\sqrt{5476639}-9268}{14}$$

Divide -9268+4\sqrt{5476639} by 14.

$$x=\frac{2\sqrt{5476639}}{7}-662$$

Now solve the equation x=\frac{-9268±4\sqrt{5476639}}{14} when ± is minus. Subtract 4\sqrt{5476639} from -9268.

$$x=\frac{-4\sqrt{5476639}-9268}{14}$$

Divide -9268-4\sqrt{5476639} by 14.

$$x=-\frac{2\sqrt{5476639}}{7}-662$$

The equation is now solved.

$$x=\frac{2\sqrt{5476639}}{7}-662 x=-\frac{2\sqrt{5476639}}{7}-662$$

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.

$$7x^{2}+9268x-61800=0$$

Add 61800 to both sides of the equation.

$$7x^{2}+9268x-61800-\left(-61800\right)=-\left(-61800\right)$$

Subtracting -61800 from itself leaves 0.

$$7x^{2}+9268x=-\left(-61800\right)$$

Subtract -61800 from 0.

$$7x^{2}+9268x=61800$$

Divide both sides by 7.

$$\frac{7x^{2}+9268x}{7}=\frac{61800}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{9268}{7}x=\frac{61800}{7}$$

Divide 9268 by 7.

$$x^{2}+1324x=\frac{61800}{7}$$

Divide 1324, the coefficient of the x term, by 2 to get 662. Then add the square of 662 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+1324x+662^{2}=\frac{61800}{7}+662^{2}$$

Square 662.

$$x^{2}+1324x+438244=\frac{61800}{7}+438244$$

Add \frac{61800}{7} to 438244.

$$x^{2}+1324x+438244=\frac{3129508}{7}$$

Factor x^{2}+1324x+438244. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+662\right)^{2}=\frac{3129508}{7}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+662\right)^{2}}=\sqrt{\frac{3129508}{7}}$$

Simplify.

$$x+662=\frac{2\sqrt{5476639}}{7} x+662=-\frac{2\sqrt{5476639}}{7}$$

Subtract 662 from both sides of the equation.

$$x=\frac{2\sqrt{5476639}}{7}-662 x=-\frac{2\sqrt{5476639}}{7}-662$$