Solve x^2+4x=9(3/4)

$$x ^ { 2 } + 4 x = 9 ( 3 / 4 )$$

Solve for x

$$x=\frac{\sqrt{43}}{2}-2\approx 1.278719262$$

$$x=-\frac{\sqrt{43}}{2}-2\approx -5.278719262$$

Steps Using the Quadratic Formula

Multiply 9 and \frac{3}{4} to get \frac{27}{4}.

$$x^{2}+4x=\frac{27}{4}$$

Subtract \frac{27}{4} from both sides.

$$x^{2}+4x-\frac{27}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -\frac{27}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-\frac{27}{4}\right)}}{2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-\frac{27}{4}\right)}}{2}$$

Multiply -4 times -\frac{27}{4}.

$$x=\frac{-4±\sqrt{16+27}}{2}$$

Add 16 to 27.

$$x=\frac{-4±\sqrt{43}}{2}$$

Now solve the equation x=\frac{-4±\sqrt{43}}{2} when ± is plus. Add -4 to \sqrt{43}.

$$x=\frac{\sqrt{43}-4}{2}$$

Divide -4+\sqrt{43} by 2.

$$x=\frac{\sqrt{43}}{2}-2$$

Now solve the equation x=\frac{-4±\sqrt{43}}{2} when ± is minus. Subtract \sqrt{43} from -4.

$$x=\frac{-\sqrt{43}-4}{2}$$

Divide -4-\sqrt{43} by 2.

$$x=-\frac{\sqrt{43}}{2}-2$$

The equation is now solved.

$$x=\frac{\sqrt{43}}{2}-2 x=-\frac{\sqrt{43}}{2}-2$$

Steps for Completing the Square

Multiply 9 and \frac{3}{4} to get \frac{27}{4}.

$$x^{2}+4x=\frac{27}{4}$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=\frac{27}{4}+2^{2}$$

Square 2.

$$x^{2}+4x+4=\frac{27}{4}+4$$

Add \frac{27}{4} to 4.

$$x^{2}+4x+4=\frac{43}{4}$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=\frac{43}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{\frac{43}{4}}$$

Simplify.

$$x+2=\frac{\sqrt{43}}{2} x+2=-\frac{\sqrt{43}}{2}$$

Subtract 2 from both sides of the equation.

$$x=\frac{\sqrt{43}}{2}-2 x=-\frac{\sqrt{43}}{2}-2$$