Solve x^2+5x+4x=45

$${ x }^{ 2 } +5x+4x=45$$

Solve for x

$$x = \frac{3 \sqrt{29} - 9}{2} \approx 3.577747211$$

$$x=\frac{-3\sqrt{29}-9}{2}\approx -12.577747211$$

Steps Using the Quadratic Formula

Combine 5x and 4x to get 9x.

$$x^{2}+9x=45$$

Subtract 45 from both sides.

$$x^{2}+9x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-45\right)}}{2}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-45\right)}}{2}$$

Multiply -4 times -45.

$$x=\frac{-9±\sqrt{81+180}}{2}$$

Add 81 to 180.

$$x=\frac{-9±\sqrt{261}}{2}$$

Take the square root of 261.

$$x=\frac{-9±3\sqrt{29}}{2}$$

Now solve the equation x=\frac{-9±3\sqrt{29}}{2} when ± is plus. Add -9 to 3\sqrt{29}.

$$x=\frac{3\sqrt{29}-9}{2}$$

Now solve the equation x=\frac{-9±3\sqrt{29}}{2} when ± is minus. Subtract 3\sqrt{29} from -9.

$$x=\frac{-3\sqrt{29}-9}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{29}-9}{2} x=\frac{-3\sqrt{29}-9}{2}$$

Steps for Completing the Square

Combine 5x and 4x to get 9x.

$$x^{2}+9x=45$$

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=45+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=45+\frac{81}{4}$$

Add 45 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{261}{4}$$

Factor x^{2}+9x+\frac{81}{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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{261}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{261}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{3\sqrt{29}}{2} x+\frac{9}{2}=-\frac{3\sqrt{29}}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=\frac{3\sqrt{29}-9}{2} x=\frac{-3\sqrt{29}-9}{2}$$