Solve 5x-2-6x^2+4x-3x^2+x-30=0

$$5x-2-6 { x }^{ 2 } +4x-3 { x }^{ 2 } +x-30=0$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{263}i+5}{9}\approx 0.555555556-1.801919416i$$

$$x=\frac{5+\sqrt{263}i}{9}\approx 0.555555556+1.801919416i$$

Steps Using the Quadratic Formula

Combine 5x and 4x to get 9x.

$$9x-2-6x^{2}-3x^{2}+x-30=0$$

Combine -6x^{2} and -3x^{2} to get -9x^{2}.

$$9x-2-9x^{2}+x-30=0$$

Combine 9x and x to get 10x.

$$10x-2-9x^{2}-30=0$$

Subtract 30 from -2 to get -32.

$$10x-32-9x^{2}=0$$

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.

$$-9x^{2}+10x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 10 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-9\right)\left(-32\right)}}{2\left(-9\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-9\right)\left(-32\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-10±\sqrt{100+36\left(-32\right)}}{2\left(-9\right)}$$

Multiply 36 times -32.

$$x=\frac{-10±\sqrt{100-1152}}{2\left(-9\right)}$$

Add 100 to -1152.

$$x=\frac{-10±\sqrt{-1052}}{2\left(-9\right)}$$

Take the square root of -1052.

$$x=\frac{-10±2\sqrt{263}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-10±2\sqrt{263}i}{-18}$$

Now solve the equation x=\frac{-10±2\sqrt{263}i}{-18} when ± is plus. Add -10 to 2i\sqrt{263}.

$$x=\frac{-10+2\sqrt{263}i}{-18}$$

Divide -10+2i\sqrt{263} by -18.

$$x=\frac{-\sqrt{263}i+5}{9}$$

Now solve the equation x=\frac{-10±2\sqrt{263}i}{-18} when ± is minus. Subtract 2i\sqrt{263} from -10.

$$x=\frac{-2\sqrt{263}i-10}{-18}$$

Divide -10-2i\sqrt{263} by -18.

$$x=\frac{5+\sqrt{263}i}{9}$$

The equation is now solved.

$$x=\frac{-\sqrt{263}i+5}{9} x=\frac{5+\sqrt{263}i}{9}$$

Steps for Completing the Square

Combine 5x and 4x to get 9x.

$$9x-2-6x^{2}-3x^{2}+x-30=0$$

Combine -6x^{2} and -3x^{2} to get -9x^{2}.

$$9x-2-9x^{2}+x-30=0$$

Combine 9x and x to get 10x.

$$10x-2-9x^{2}-30=0$$

Subtract 30 from -2 to get -32.

$$10x-32-9x^{2}=0$$

Add 32 to both sides. Anything plus zero gives itself.

$$10x-9x^{2}=32$$

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.

$$-9x^{2}+10x=32$$

Divide both sides by -9.

$$\frac{-9x^{2}+10x}{-9}=\frac{32}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{10}{-9}x=\frac{32}{-9}$$

Divide 10 by -9.

$$x^{2}-\frac{10}{9}x=\frac{32}{-9}$$

Divide 32 by -9.

$$x^{2}-\frac{10}{9}x=-\frac{32}{9}$$

Divide -\frac{10}{9}, the coefficient of the x term, by 2 to get -\frac{5}{9}. Then add the square of -\frac{5}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{10}{9}x+\left(-\frac{5}{9}\right)^{2}=-\frac{32}{9}+\left(-\frac{5}{9}\right)^{2}$$

Square -\frac{5}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{10}{9}x+\frac{25}{81}=-\frac{32}{9}+\frac{25}{81}$$

Add -\frac{32}{9} to \frac{25}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{10}{9}x+\frac{25}{81}=-\frac{263}{81}$$

Factor x^{2}-\frac{10}{9}x+\frac{25}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{9}\right)^{2}=-\frac{263}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{9}\right)^{2}}=\sqrt{-\frac{263}{81}}$$

Simplify.

$$x-\frac{5}{9}=\frac{\sqrt{263}i}{9} x-\frac{5}{9}=-\frac{\sqrt{263}i}{9}$$

Add \frac{5}{9} to both sides of the equation.

$$x=\frac{5+\sqrt{263}i}{9} x=\frac{-\sqrt{263}i+5}{9}$$