Solve x^2+34x=3550*2

$${ x }^{ 2 } +34x=3550 \times 2$$

Solve for x

$$x=3\sqrt{821}-17\approx 68.959292691$$

$$x=-3\sqrt{821}-17\approx -102.959292691$$

Steps Using the Quadratic Formula

Multiply 3550 and 2 to get 7100.

$$x^{2}+34x=7100$$

Subtract 7100 from both sides.

$$x^{2}+34x-7100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 34 for b, and -7100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-7100\right)}}{2}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-7100\right)}}{2}$$

Multiply -4 times -7100.

$$x=\frac{-34±\sqrt{1156+28400}}{2}$$

Add 1156 to 28400.

$$x=\frac{-34±\sqrt{29556}}{2}$$

Take the square root of 29556.

$$x=\frac{-34±6\sqrt{821}}{2}$$

Now solve the equation x=\frac{-34±6\sqrt{821}}{2} when ± is plus. Add -34 to 6\sqrt{821}.

$$x=\frac{6\sqrt{821}-34}{2}$$

Divide -34+6\sqrt{821} by 2.

$$x=3\sqrt{821}-17$$

Now solve the equation x=\frac{-34±6\sqrt{821}}{2} when ± is minus. Subtract 6\sqrt{821} from -34.

$$x=\frac{-6\sqrt{821}-34}{2}$$

Divide -34-6\sqrt{821} by 2.

$$x=-3\sqrt{821}-17$$

The equation is now solved.

$$x=3\sqrt{821}-17 x=-3\sqrt{821}-17$$

Steps for Completing the Square

Multiply 3550 and 2 to get 7100.

$$x^{2}+34x=7100$$

Divide 34, the coefficient of the x term, by 2 to get 17. Then add the square of 17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+34x+17^{2}=7100+17^{2}$$

Square 17.

$$x^{2}+34x+289=7100+289$$

Add 7100 to 289.

$$x^{2}+34x+289=7389$$

Factor x^{2}+34x+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+17\right)^{2}=7389$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+17\right)^{2}}=\sqrt{7389}$$

Simplify.

$$x+17=3\sqrt{821} x+17=-3\sqrt{821}$$

Subtract 17 from both sides of the equation.

$$x=3\sqrt{821}-17 x=-3\sqrt{821}-17$$