Solve (3-6x)-17=(4-3x)*(5-4x)-2

$$( 3 - 6 x ) - 17 = ( 4 - 3 x ) \cdot ( 5 - 4 x ) - 2$$

Solve for x (complex solution)

$$x=\frac{25+\sqrt{911}i}{24}\approx 1.041666667+1.257615689i$$

$$x=\frac{-\sqrt{911}i+25}{24}\approx 1.041666667-1.257615689i$$

Steps Using the Quadratic Formula

Subtract 17 from 3 to get -14.

$$-14-6x=\left(4-3x\right)\left(5-4x\right)-2$$

Use the distributive property to multiply 4-3x by 5-4x and combine like terms.

$$-14-6x=20-31x+12x^{2}-2$$

Subtract 2 from 20 to get 18.

$$-14-6x=18-31x+12x^{2}$$

Subtract 18 from both sides.

$$-14-6x-18=-31x+12x^{2}$$

Subtract 18 from -14 to get -32.

$$-32-6x=-31x+12x^{2}$$

Add 31x to both sides.

$$-32-6x+31x=12x^{2}$$

Combine -6x and 31x to get 25x.

$$-32+25x=12x^{2}$$

Subtract 12x^{2} from both sides.

$$-32+25x-12x^{2}=0$$

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.

$$-12x^{2}+25x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, 25 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-12\right)\left(-32\right)}}{2\left(-12\right)}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\left(-12\right)\left(-32\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-25±\sqrt{625+48\left(-32\right)}}{2\left(-12\right)}$$

Multiply 48 times -32.

$$x=\frac{-25±\sqrt{625-1536}}{2\left(-12\right)}$$

Add 625 to -1536.

$$x=\frac{-25±\sqrt{-911}}{2\left(-12\right)}$$

Take the square root of -911.

$$x=\frac{-25±\sqrt{911}i}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-25±\sqrt{911}i}{-24}$$

Now solve the equation x=\frac{-25±\sqrt{911}i}{-24} when ± is plus. Add -25 to i\sqrt{911}.

$$x=\frac{-25+\sqrt{911}i}{-24}$$

Divide -25+i\sqrt{911} by -24.

$$x=\frac{-\sqrt{911}i+25}{24}$$

Now solve the equation x=\frac{-25±\sqrt{911}i}{-24} when ± is minus. Subtract i\sqrt{911} from -25.

$$x=\frac{-\sqrt{911}i-25}{-24}$$

Divide -25-i\sqrt{911} by -24.

$$x=\frac{25+\sqrt{911}i}{24}$$

The equation is now solved.

$$x=\frac{-\sqrt{911}i+25}{24} x=\frac{25+\sqrt{911}i}{24}$$

Steps for Completing the Square

Subtract 17 from 3 to get -14.

$$-14-6x=\left(4-3x\right)\left(5-4x\right)-2$$

Use the distributive property to multiply 4-3x by 5-4x and combine like terms.

$$-14-6x=20-31x+12x^{2}-2$$

Subtract 2 from 20 to get 18.

$$-14-6x=18-31x+12x^{2}$$

Add 31x to both sides.

$$-14-6x+31x=18+12x^{2}$$

Combine -6x and 31x to get 25x.

$$-14+25x=18+12x^{2}$$

Subtract 12x^{2} from both sides.

$$-14+25x-12x^{2}=18$$

Add 14 to both sides.

$$25x-12x^{2}=18+14$$

Add 18 and 14 to get 32.

$$25x-12x^{2}=32$$

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.

$$-12x^{2}+25x=32$$

Divide both sides by -12.

$$\frac{-12x^{2}+25x}{-12}=\frac{32}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$x^{2}+\frac{25}{-12}x=\frac{32}{-12}$$

Divide 25 by -12.

$$x^{2}-\frac{25}{12}x=\frac{32}{-12}$$

Reduce the fraction \frac{32}{-12} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{25}{12}x=-\frac{8}{3}$$

Divide -\frac{25}{12}, the coefficient of the x term, by 2 to get -\frac{25}{24}. Then add the square of -\frac{25}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{12}x+\left(-\frac{25}{24}\right)^{2}=-\frac{8}{3}+\left(-\frac{25}{24}\right)^{2}$$

Square -\frac{25}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{12}x+\frac{625}{576}=-\frac{8}{3}+\frac{625}{576}$$

Add -\frac{8}{3} to \frac{625}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{12}x+\frac{625}{576}=-\frac{911}{576}$$

Factor x^{2}-\frac{25}{12}x+\frac{625}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{24}\right)^{2}=-\frac{911}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{24}\right)^{2}}=\sqrt{-\frac{911}{576}}$$

Simplify.

$$x-\frac{25}{24}=\frac{\sqrt{911}i}{24} x-\frac{25}{24}=-\frac{\sqrt{911}i}{24}$$

Add \frac{25}{24} to both sides of the equation.

$$x=\frac{25+\sqrt{911}i}{24} x=\frac{-\sqrt{911}i+25}{24}$$