Solve x^2+340x+4516=0

$${ x }^{ 2 } +340x+4516=0$$

Solve for x

$$x=8\sqrt{381}-170\approx -13.846229632$$

$$x=-8\sqrt{381}-170\approx -326.153770368$$

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.

$$x^{2}+340x+4516=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 340 for b, and 4516 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-340±\sqrt{340^{2}-4\times 4516}}{2}$$

Square 340.

$$x=\frac{-340±\sqrt{115600-4\times 4516}}{2}$$

Multiply -4 times 4516.

$$x=\frac{-340±\sqrt{115600-18064}}{2}$$

Add 115600 to -18064.

$$x=\frac{-340±\sqrt{97536}}{2}$$

Take the square root of 97536.

$$x=\frac{-340±16\sqrt{381}}{2}$$

Now solve the equation x=\frac{-340±16\sqrt{381}}{2} when ± is plus. Add -340 to 16\sqrt{381}.

$$x=\frac{16\sqrt{381}-340}{2}$$

Divide -340+16\sqrt{381} by 2.

$$x=8\sqrt{381}-170$$

Now solve the equation x=\frac{-340±16\sqrt{381}}{2} when ± is minus. Subtract 16\sqrt{381} from -340.

$$x=\frac{-16\sqrt{381}-340}{2}$$

Divide -340-16\sqrt{381} by 2.

$$x=-8\sqrt{381}-170$$

The equation is now solved.

$$x=8\sqrt{381}-170 x=-8\sqrt{381}-170$$

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.

$$x^{2}+340x+4516=0$$

Subtract 4516 from both sides of the equation.

$$x^{2}+340x+4516-4516=-4516$$

Subtracting 4516 from itself leaves 0.

$$x^{2}+340x=-4516$$

Divide 340, the coefficient of the x term, by 2 to get 170. Then add the square of 170 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+340x+170^{2}=-4516+170^{2}$$

Square 170.

$$x^{2}+340x+28900=-4516+28900$$

Add -4516 to 28900.

$$x^{2}+340x+28900=24384$$

Factor x^{2}+340x+28900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+170\right)^{2}=24384$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+170\right)^{2}}=\sqrt{24384}$$

Simplify.

$$x+170=8\sqrt{381} x+170=-8\sqrt{381}$$

Subtract 170 from both sides of the equation.

$$x=8\sqrt{381}-170 x=-8\sqrt{381}-170$$