Solve x^2+17.38x-20772=0

$${ x }^{ 2 } +17.38x-20772=0$$

Solve for x

$$x = \frac{\sqrt{208475161} - 869}{100} \approx 135.696689483$$

$$x=\frac{-\sqrt{208475161}-869}{100}\approx -153.076689483$$

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.

$$x^{2}+17.38x-20772=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 17.38 for b, and -20772 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17.38±\sqrt{17.38^{2}-4\left(-20772\right)}}{2}$$

Square 17.38 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-17.38±\sqrt{302.0644-4\left(-20772\right)}}{2}$$

Multiply -4 times -20772.

$$x=\frac{-17.38±\sqrt{302.0644+83088}}{2}$$

Add 302.0644 to 83088.

$$x=\frac{-17.38±\sqrt{83390.0644}}{2}$$

Take the square root of 83390.0644.

$$x=\frac{-17.38±\frac{\sqrt{208475161}}{50}}{2}$$

Now solve the equation x=\frac{-17.38±\frac{\sqrt{208475161}}{50}}{2} when ± is plus. Add -17.38 to \frac{\sqrt{208475161}}{50}.

$$x=\frac{\sqrt{208475161}-869}{2\times 50}$$

Divide \frac{-869+\sqrt{208475161}}{50} by 2.

$$x=\frac{\sqrt{208475161}-869}{100}$$

Now solve the equation x=\frac{-17.38±\frac{\sqrt{208475161}}{50}}{2} when ± is minus. Subtract \frac{\sqrt{208475161}}{50} from -17.38.

$$x=\frac{-\sqrt{208475161}-869}{2\times 50}$$

Divide \frac{-869-\sqrt{208475161}}{50} by 2.

$$x=\frac{-\sqrt{208475161}-869}{100}$$

The equation is now solved.

$$x=\frac{\sqrt{208475161}-869}{100} x=\frac{-\sqrt{208475161}-869}{100}$$

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.

$$x^{2}+17.38x-20772=0$$

Add 20772 to both sides of the equation.

$$x^{2}+17.38x-20772-\left(-20772\right)=-\left(-20772\right)$$

Subtracting -20772 from itself leaves 0.

$$x^{2}+17.38x=-\left(-20772\right)$$

Subtract -20772 from 0.

$$x^{2}+17.38x=20772$$

Divide 17.38, the coefficient of the x term, by 2 to get 8.69. Then add the square of 8.69 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+17.38x+8.69^{2}=20772+8.69^{2}$$

Square 8.69 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+17.38x+75.5161=20772+75.5161$$

Add 20772 to 75.5161.

$$x^{2}+17.38x+75.5161=20847.5161$$

Factor x^{2}+17.38x+75.5161. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+8.69\right)^{2}=20847.5161$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8.69\right)^{2}}=\sqrt{20847.5161}$$

Simplify.

$$x+8.69=\frac{\sqrt{208475161}}{100} x+8.69=-\frac{\sqrt{208475161}}{100}$$

Subtract 8.69 from both sides of the equation.

$$x=\frac{\sqrt{208475161}-869}{100} x=\frac{-\sqrt{208475161}-869}{100}$$