Solve (3x-9)x=5670

$$( 3 x - 9 ) x = 5670$$

Solve for x

$$x=-42$$

$$x=45$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x-9 by x.

$$3x^{2}-9x=5670$$

Subtract 5670 from both sides.

$$3x^{2}-9x-5670=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -9 for b, and -5670 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 3\left(-5670\right)}}{2\times 3}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 3\left(-5670\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-9\right)±\sqrt{81-12\left(-5670\right)}}{2\times 3}$$

Multiply -12 times -5670.

$$x=\frac{-\left(-9\right)±\sqrt{81+68040}}{2\times 3}$$

Add 81 to 68040.

$$x=\frac{-\left(-9\right)±\sqrt{68121}}{2\times 3}$$

Take the square root of 68121.

$$x=\frac{-\left(-9\right)±261}{2\times 3}$$

The opposite of -9 is 9.

$$x=\frac{9±261}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{9±261}{6}$$

Now solve the equation x=\frac{9±261}{6} when ± is plus. Add 9 to 261.

$$x=\frac{270}{6}$$

Divide 270 by 6.

$$x=45$$

Now solve the equation x=\frac{9±261}{6} when ± is minus. Subtract 261 from 9.

$$x=-\frac{252}{6}$$

Divide -252 by 6.

$$x=-42$$

The equation is now solved.

$$x=45 x=-42$$

Steps for Completing the Square

Use the distributive property to multiply 3x-9 by x.

$$3x^{2}-9x=5670$$

Divide both sides by 3.

$$\frac{3x^{2}-9x}{3}=\frac{5670}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{9}{3}\right)x=\frac{5670}{3}$$

Divide -9 by 3.

$$x^{2}-3x=\frac{5670}{3}$$

Divide 5670 by 3.

$$x^{2}-3x=1890$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=1890+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=1890+\frac{9}{4}$$

Add 1890 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{7569}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{7569}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{7569}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{87}{2} x-\frac{3}{2}=-\frac{87}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=45 x=-42$$