Solve 1601.76=x^2+5x+4

$$1601.76= { x }^{ 2 } +5x+4$$

Solve for x

$$x=\frac{\sqrt{160401}}{10}-2.5\approx 37.550093633$$

$$x=-\frac{\sqrt{160401}}{10}-2.5\approx -42.550093633$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+5x+4=1601.76$$

Subtract 1601.76 from both sides.

$$x^{2}+5x+4-1601.76=0$$

Subtract 1601.76 from 4 to get -1597.76.

$$x^{2}+5x-1597.76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and -1597.76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1597.76\right)}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1597.76\right)}}{2}$$

Multiply -4 times -1597.76.

$$x=\frac{-5±\sqrt{25+6391.04}}{2}$$

Add 25 to 6391.04.

$$x=\frac{-5±\sqrt{6416.04}}{2}$$

Take the square root of 6416.04.

$$x=\frac{-5±\frac{\sqrt{160401}}{5}}{2}$$

Now solve the equation x=\frac{-5±\frac{\sqrt{160401}}{5}}{2} when ± is plus. Add -5 to \frac{\sqrt{160401}}{5}.

$$x=\frac{\frac{\sqrt{160401}}{5}-5}{2}$$

Divide -5+\frac{\sqrt{160401}}{5} by 2.

$$x=\frac{\sqrt{160401}}{10}-\frac{5}{2}$$

Now solve the equation x=\frac{-5±\frac{\sqrt{160401}}{5}}{2} when ± is minus. Subtract \frac{\sqrt{160401}}{5} from -5.

$$x=\frac{-\frac{\sqrt{160401}}{5}-5}{2}$$

Divide -5-\frac{\sqrt{160401}}{5} by 2.

$$x=-\frac{\sqrt{160401}}{10}-\frac{5}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{160401}}{10}-\frac{5}{2} x=-\frac{\sqrt{160401}}{10}-\frac{5}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+5x+4=1601.76$$

Subtract 4 from both sides.

$$x^{2}+5x=1601.76-4$$

Subtract 4 from 1601.76 to get 1597.76.

$$x^{2}+5x=1597.76$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=1597.76+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=1597.76+\frac{25}{4}$$

Add 1597.76 to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+5x+\frac{25}{4}=\frac{160401}{100}$$

Factor x^{2}+5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{160401}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{160401}{100}}$$

Simplify.

$$x+\frac{5}{2}=\frac{\sqrt{160401}}{10} x+\frac{5}{2}=-\frac{\sqrt{160401}}{10}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=\frac{\sqrt{160401}}{10}-\frac{5}{2} x=-\frac{\sqrt{160401}}{10}-\frac{5}{2}$$