Solve x:3x+4=9x+3

$$x : 3 x + 4 = 9 x + 3$$

Solve for x

$$x = \frac{\sqrt{717} + 27}{2} \approx 26.888427839$$

$$x=\frac{27-\sqrt{717}}{2}\approx 0.111572161$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$xx+12=27x+9$$

Multiply x and x to get x^{2}.

$$x^{2}+12=27x+9$$

Subtract 27x from both sides.

$$x^{2}+12-27x=9$$

Subtract 9 from both sides.

$$x^{2}+12-27x-9=0$$

Subtract 9 from 12 to get 3.

$$x^{2}+3-27x=0$$

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.

$$x^{2}-27x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -27 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 3}}{2}$$

Square -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\times 3}}{2}$$

Multiply -4 times 3.

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

Add 729 to -12.

$$x=\frac{-\left(-27\right)±\sqrt{717}}{2}$$

The opposite of -27 is 27.

$$x=\frac{27±\sqrt{717}}{2}$$

Now solve the equation x=\frac{27±\sqrt{717}}{2} when ± is plus. Add 27 to \sqrt{717}.

$$x=\frac{\sqrt{717}+27}{2}$$

Now solve the equation x=\frac{27±\sqrt{717}}{2} when ± is minus. Subtract \sqrt{717} from 27.

$$x=\frac{27-\sqrt{717}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{717}+27}{2} x=\frac{27-\sqrt{717}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$xx+12=27x+9$$

Multiply x and x to get x^{2}.

$$x^{2}+12=27x+9$$

Subtract 27x from both sides.

$$x^{2}+12-27x=9$$

Subtract 12 from both sides.

$$x^{2}-27x=9-12$$

Subtract 12 from 9 to get -3.

$$x^{2}-27x=-3$$

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.

$$x^{2}-27x+\left(-\frac{27}{2}\right)^{2}=-3+\left(-\frac{27}{2}\right)^{2}$$

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

$$x^{2}-27x+\frac{729}{4}=-3+\frac{729}{4}$$

Add -3 to \frac{729}{4}.

$$x^{2}-27x+\frac{729}{4}=\frac{717}{4}$$

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}.

$$\left(x-\frac{27}{2}\right)^{2}=\frac{717}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{2}\right)^{2}}=\sqrt{\frac{717}{4}}$$

Simplify.

$$x-\frac{27}{2}=\frac{\sqrt{717}}{2} x-\frac{27}{2}=-\frac{\sqrt{717}}{2}$$

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

$$x=\frac{\sqrt{717}+27}{2} x=\frac{27-\sqrt{717}}{2}$$