Solve (x+1)*(3x)+2=1064

$$( x + 1 ) \cdot ( 3 x ) + 2 = 1064$$

Solve for x

$$x = \frac{\sqrt{1417} - 1}{2} \approx 18.321530225$$

$$x=\frac{-\sqrt{1417}-1}{2}\approx -19.321530225$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+1 by 3.

$$\left(3x+3\right)x+2=1064$$

Use the distributive property to multiply 3x+3 by x.

$$3x^{2}+3x+2=1064$$

Subtract 1064 from both sides.

$$3x^{2}+3x+2-1064=0$$

Subtract 1064 from 2 to get -1062.

$$3x^{2}+3x-1062=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 3 for b, and -1062 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 3\left(-1062\right)}}{2\times 3}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 3\left(-1062\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-3±\sqrt{9-12\left(-1062\right)}}{2\times 3}$$

Multiply -12 times -1062.

$$x=\frac{-3±\sqrt{9+12744}}{2\times 3}$$

Add 9 to 12744.

$$x=\frac{-3±\sqrt{12753}}{2\times 3}$$

Take the square root of 12753.

$$x=\frac{-3±3\sqrt{1417}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-3±3\sqrt{1417}}{6}$$

Now solve the equation x=\frac{-3±3\sqrt{1417}}{6} when ± is plus. Add -3 to 3\sqrt{1417}.

$$x=\frac{3\sqrt{1417}-3}{6}$$

Divide -3+3\sqrt{1417} by 6.

$$x=\frac{\sqrt{1417}-1}{2}$$

Now solve the equation x=\frac{-3±3\sqrt{1417}}{6} when ± is minus. Subtract 3\sqrt{1417} from -3.

$$x=\frac{-3\sqrt{1417}-3}{6}$$

Divide -3-3\sqrt{1417} by 6.

$$x=\frac{-\sqrt{1417}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1417}-1}{2} x=\frac{-\sqrt{1417}-1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x+1 by 3.

$$\left(3x+3\right)x+2=1064$$

Use the distributive property to multiply 3x+3 by x.

$$3x^{2}+3x+2=1064$$

Subtract 2 from both sides.

$$3x^{2}+3x=1064-2$$

Subtract 2 from 1064 to get 1062.

$$3x^{2}+3x=1062$$

Divide both sides by 3.

$$\frac{3x^{2}+3x}{3}=\frac{1062}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{3}{3}x=\frac{1062}{3}$$

Divide 3 by 3.

$$x^{2}+x=\frac{1062}{3}$$

Divide 1062 by 3.

$$x^{2}+x=354$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=354+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=354+\frac{1}{4}$$

Add 354 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{1417}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{1417}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{1417}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{1417}}{2} x+\frac{1}{2}=-\frac{\sqrt{1417}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{1417}-1}{2} x=\frac{-\sqrt{1417}-1}{2}$$