Solve x^2-27x/4=3

$$\frac { x ^ { 2 } - 27 x } { 4 } = 3$$

Solve for x

$$x = \frac{\sqrt{777} + 27}{2} \approx 27.437359865$$

$$x=\frac{27-\sqrt{777}}{2}\approx -0.437359865$$

Steps Using the Quadratic Formula

Multiply both sides by 4.

$$x^{2}-27x=3\times 4$$

Multiply 3 and 4 to get 12.

$$x^{2}-27x=12$$

Subtract 12 from both sides.

$$x^{2}-27x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -27 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\left(-12\right)}}{2}$$

Square -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\left(-12\right)}}{2}$$

Multiply -4 times -12.

$$x=\frac{-\left(-27\right)±\sqrt{729+48}}{2}$$

Add 729 to 48.

$$x=\frac{-\left(-27\right)±\sqrt{777}}{2}$$

The opposite of -27 is 27.

$$x=\frac{27±\sqrt{777}}{2}$$

Now solve the equation x=\frac{27±\sqrt{777}}{2} when ± is plus. Add 27 to \sqrt{777}.

$$x=\frac{\sqrt{777}+27}{2}$$

Now solve the equation x=\frac{27±\sqrt{777}}{2} when ± is minus. Subtract \sqrt{777} from 27.

$$x=\frac{27-\sqrt{777}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{777}+27}{2} x=\frac{27-\sqrt{777}}{2}$$

Steps for Completing the Square

Multiply both sides by 4.

$$x^{2}-27x=3\times 4$$

Multiply 3 and 4 to get 12.

$$x^{2}-27x=12$$

Divide -27, the coefficient of the x term, by 2 to get -\frac{27}{2}. Then add the square of -\frac{27}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-27x+\left(-\frac{27}{2}\right)^{2}=12+\left(-\frac{27}{2}\right)^{2}$$

Square -\frac{27}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-27x+\frac{729}{4}=12+\frac{729}{4}$$

Add 12 to \frac{729}{4}.

$$x^{2}-27x+\frac{729}{4}=\frac{777}{4}$$

Factor x^{2}-27x+\frac{729}{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}.

$$\left(x-\frac{27}{2}\right)^{2}=\frac{777}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{2}\right)^{2}}=\sqrt{\frac{777}{4}}$$

Simplify.

$$x-\frac{27}{2}=\frac{\sqrt{777}}{2} x-\frac{27}{2}=-\frac{\sqrt{777}}{2}$$

Add \frac{27}{2} to both sides of the equation.

$$x=\frac{\sqrt{777}+27}{2} x=\frac{27-\sqrt{777}}{2}$$