Solve -2x^2+6x-10=-3x^2

$$- 2 x ^ { 2 } + 6 x - 10 = - 3 x ^ { 2 }$$

Solve for x (complex solution)

$$x=\sqrt{19}-3\approx 1.358898944$$

$$x=-\left(\sqrt{19}+3\right)\approx -7.358898944$$

Steps Using the Quadratic Formula

Add 3x^{2} to both sides.

$$-2x^{2}+6x-10+3x^{2}=0$$

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

$$x^{2}+6x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-10\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-10\right)}}{2}$$

Multiply -4 times -10.

$$x=\frac{-6±\sqrt{36+40}}{2}$$

Add 36 to 40.

$$x=\frac{-6±\sqrt{76}}{2}$$

Take the square root of 76.

$$x=\frac{-6±2\sqrt{19}}{2}$$

Now solve the equation x=\frac{-6±2\sqrt{19}}{2} when ± is plus. Add -6 to 2\sqrt{19}.

$$x=\frac{2\sqrt{19}-6}{2}$$

Divide -6+2\sqrt{19} by 2.

$$x=\sqrt{19}-3$$

Now solve the equation x=\frac{-6±2\sqrt{19}}{2} when ± is minus. Subtract 2\sqrt{19} from -6.

$$x=\frac{-2\sqrt{19}-6}{2}$$

Divide -6-2\sqrt{19} by 2.

$$x=-\sqrt{19}-3$$

The equation is now solved.

$$x=\sqrt{19}-3 x=-\sqrt{19}-3$$

Steps for Completing the Square

Add 3x^{2} to both sides.

$$-2x^{2}+6x-10+3x^{2}=0$$

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

$$x^{2}+6x-10=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$x^{2}+6x=10$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=10+3^{2}$$

Square 3.

$$x^{2}+6x+9=10+9$$

Add 10 to 9.

$$x^{2}+6x+9=19$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=19$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{19}$$

Simplify.

$$x+3=\sqrt{19} x+3=-\sqrt{19}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{19}-3 x=-\sqrt{19}-3$$

Solve for x

$$x=\sqrt{19}-3\approx 1.358898944$$

$$x=-\sqrt{19}-3\approx -7.358898944$$

Steps Using the Quadratic Formula

Add 3x^{2} to both sides.

$$-2x^{2}+6x-10+3x^{2}=0$$

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

$$x^{2}+6x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-10\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-10\right)}}{2}$$

Multiply -4 times -10.

$$x=\frac{-6±\sqrt{36+40}}{2}$$

Add 36 to 40.

$$x=\frac{-6±\sqrt{76}}{2}$$

Take the square root of 76.

$$x=\frac{-6±2\sqrt{19}}{2}$$

Now solve the equation x=\frac{-6±2\sqrt{19}}{2} when ± is plus. Add -6 to 2\sqrt{19}.

$$x=\frac{2\sqrt{19}-6}{2}$$

Divide -6+2\sqrt{19} by 2.

$$x=\sqrt{19}-3$$

Now solve the equation x=\frac{-6±2\sqrt{19}}{2} when ± is minus. Subtract 2\sqrt{19} from -6.

$$x=\frac{-2\sqrt{19}-6}{2}$$

Divide -6-2\sqrt{19} by 2.

$$x=-\sqrt{19}-3$$

The equation is now solved.

$$x=\sqrt{19}-3 x=-\sqrt{19}-3$$

Steps for Completing the Square

Add 3x^{2} to both sides.

$$-2x^{2}+6x-10+3x^{2}=0$$

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

$$x^{2}+6x-10=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$x^{2}+6x=10$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=10+3^{2}$$

Square 3.

$$x^{2}+6x+9=10+9$$

Add 10 to 9.

$$x^{2}+6x+9=19$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=19$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{19}$$

Simplify.

$$x+3=\sqrt{19} x+3=-\sqrt{19}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{19}-3 x=-\sqrt{19}-3$$