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

$$3 \times ( x - 3 ) = 2 ( x - 1 ) ( x + 1 )$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{47}i+3}{4}\approx 0.75-1.71391365i$$

$$x=\frac{3+\sqrt{47}i}{4}\approx 0.75+1.71391365i$$

Steps Using the Quadratic Formula

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

$$3x-9=2\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2 by x-1.

$$3x-9=\left(2x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 2x-2 by x+1 and combine like terms.

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

Subtract 2x^{2} from both sides.

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

Add 2 to both sides.

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

Add -9 and 2 to get -7.

$$3x-7-2x^{2}=0$$

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.

$$-2x^{2}+3x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 3 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-2\right)\left(-7\right)}}{2\left(-2\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-2\right)\left(-7\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-3±\sqrt{9+8\left(-7\right)}}{2\left(-2\right)}$$

Multiply 8 times -7.

$$x=\frac{-3±\sqrt{9-56}}{2\left(-2\right)}$$

Add 9 to -56.

$$x=\frac{-3±\sqrt{-47}}{2\left(-2\right)}$$

Take the square root of -47.

$$x=\frac{-3±\sqrt{47}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-3±\sqrt{47}i}{-4}$$

Now solve the equation x=\frac{-3±\sqrt{47}i}{-4} when ± is plus. Add -3 to i\sqrt{47}.

$$x=\frac{-3+\sqrt{47}i}{-4}$$

Divide -3+i\sqrt{47} by -4.

$$x=\frac{-\sqrt{47}i+3}{4}$$

Now solve the equation x=\frac{-3±\sqrt{47}i}{-4} when ± is minus. Subtract i\sqrt{47} from -3.

$$x=\frac{-\sqrt{47}i-3}{-4}$$

Divide -3-i\sqrt{47} by -4.

$$x=\frac{3+\sqrt{47}i}{4}$$

The equation is now solved.

$$x=\frac{-\sqrt{47}i+3}{4} x=\frac{3+\sqrt{47}i}{4}$$

Steps for Completing the Square

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

$$3x-9=2\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2 by x-1.

$$3x-9=\left(2x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 2x-2 by x+1 and combine like terms.

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

Subtract 2x^{2} from both sides.

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

Add 9 to both sides.

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

Add -2 and 9 to get 7.

$$3x-2x^{2}=7$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-2x^{2}+3x=7$$

Divide both sides by -2.

$$\frac{-2x^{2}+3x}{-2}=\frac{7}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{3}{-2}x=\frac{7}{-2}$$

Divide 3 by -2.

$$x^{2}-\frac{3}{2}x=\frac{7}{-2}$$

Divide 7 by -2.

$$x^{2}-\frac{3}{2}x=-\frac{7}{2}$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{7}{2}+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{7}{2}+\frac{9}{16}$$

Add -\frac{7}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{47}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=-\frac{47}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{-\frac{47}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{47}i}{4} x-\frac{3}{4}=-\frac{\sqrt{47}i}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{3+\sqrt{47}i}{4} x=\frac{-\sqrt{47}i+3}{4}$$