Solve 2x-3x^2-4x+2x^2-3=7

$$2 x - 3 x ^ { 2 } - 4 x + 2 x ^ { 2 } - 3 = 7$$

Solve for x (complex solution)

$$x=-1-3i$$

$$x=-1+3i$$

Steps Using the Quadratic Formula

Combine 2x and -4x to get -2x.

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

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

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

Subtract 7 from both sides.

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

Subtract 7 from -3 to get -10.

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

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.

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

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

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\left(-10\right)}}{2\left(-1\right)}$$

Multiply 4 times -10.

$$x=\frac{-\left(-2\right)±\sqrt{4-40}}{2\left(-1\right)}$$

Add 4 to -40.

$$x=\frac{-\left(-2\right)±\sqrt{-36}}{2\left(-1\right)}$$

Take the square root of -36.

$$x=\frac{-\left(-2\right)±6i}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±6i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±6i}{-2}$$

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

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

Divide 2+6i by -2.

$$x=-1-3i$$

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

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

Divide 2-6i by -2.

$$x=-1+3i$$

The equation is now solved.

$$x=-1-3i x=-1+3i$$

Steps for Completing the Square

Combine 2x and -4x to get -2x.

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

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

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

Add 3 to both sides.

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

Add 7 and 3 to get 10.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}-2x}{-1}=\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{2}{-1}\right)x=\frac{10}{-1}$$

Divide -2 by -1.

$$x^{2}+2x=\frac{10}{-1}$$

Divide 10 by -1.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=-10+1^{2}$$

Square 1.

$$x^{2}+2x+1=-10+1$$

Add -10 to 1.

$$x^{2}+2x+1=-9$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=-9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-9}$$

Simplify.

$$x+1=3i x+1=-3i$$

Subtract 1 from both sides of the equation.

$$x=-1+3i x=-1-3i$$