Solve 4x^2-12x+10=3x^2-16x

$$4 x ^ { 2 } - 12 x + 10 = 3 x ^ { 2 } - 16 x$$

Solve for x (complex solution)

$$x=-2+\sqrt{6}i\approx -2+2.449489743i$$

$$x=-\sqrt{6}i-2\approx -2-2.449489743i$$

Steps Using the Quadratic Formula

Subtract 3x^{2} from both sides.

$$4x^{2}-12x+10-3x^{2}=-16x$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}-12x+10=-16x$$

Add 16x to both sides.

$$x^{2}-12x+10+16x=0$$

Combine -12x and 16x to get 4x.

$$x^{2}+4x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 10}}{2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-4±\sqrt{16-40}}{2}$$

Add 16 to -40.

$$x=\frac{-4±\sqrt{-24}}{2}$$

Take the square root of -24.

$$x=\frac{-4±2\sqrt{6}i}{2}$$

Now solve the equation x=\frac{-4±2\sqrt{6}i}{2} when ± is plus. Add -4 to 2i\sqrt{6}.

$$x=\frac{-4+2\sqrt{6}i}{2}$$

Divide -4+2i\sqrt{6} by 2.

$$x=-2+\sqrt{6}i$$

Now solve the equation x=\frac{-4±2\sqrt{6}i}{2} when ± is minus. Subtract 2i\sqrt{6} from -4.

$$x=\frac{-2\sqrt{6}i-4}{2}$$

Divide -4-2i\sqrt{6} by 2.

$$x=-\sqrt{6}i-2$$

The equation is now solved.

$$x=-2+\sqrt{6}i x=-\sqrt{6}i-2$$

Steps for Completing the Square

Subtract 3x^{2} from both sides.

$$4x^{2}-12x+10-3x^{2}=-16x$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}-12x+10=-16x$$

Add 16x to both sides.

$$x^{2}-12x+10+16x=0$$

Combine -12x and 16x to get 4x.

$$x^{2}+4x+10=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+4x=-10$$

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

$$x^{2}+4x+2^{2}=-10+2^{2}$$

Square 2.

$$x^{2}+4x+4=-10+4$$

Add -10 to 4.

$$x^{2}+4x+4=-6$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=-6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{-6}$$

Simplify.

$$x+2=\sqrt{6}i x+2=-\sqrt{6}i$$

Subtract 2 from both sides of the equation.

$$x=-2+\sqrt{6}i x=-\sqrt{6}i-2$$