Solve 2x^2+11x-23=-x+3

$$2 { x }^{ 2 } +11x-23=-x+3$$

Solve for x (complex solution)

$$x=\sqrt{22}-3\approx 1.69041576$$

$$x=-\left(\sqrt{22}+3\right)\approx -7.69041576$$

Steps Using the Quadratic Formula

Add x to both sides.

$$2x^{2}+11x-23+x=3$$

Combine 11x and x to get 12x.

$$2x^{2}+12x-23=3$$

Subtract 3 from both sides.

$$2x^{2}+12x-23-3=0$$

Subtract 3 from -23 to get -26.

$$2x^{2}+12x-26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 12 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 2\left(-26\right)}}{2\times 2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 2\left(-26\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-12±\sqrt{144-8\left(-26\right)}}{2\times 2}$$

Multiply -8 times -26.

$$x=\frac{-12±\sqrt{144+208}}{2\times 2}$$

Add 144 to 208.

$$x=\frac{-12±\sqrt{352}}{2\times 2}$$

Take the square root of 352.

$$x=\frac{-12±4\sqrt{22}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-12±4\sqrt{22}}{4}$$

Now solve the equation x=\frac{-12±4\sqrt{22}}{4} when ± is plus. Add -12 to 4\sqrt{22}.

$$x=\frac{4\sqrt{22}-12}{4}$$

Divide -12+4\sqrt{22} by 4.

$$x=\sqrt{22}-3$$

Now solve the equation x=\frac{-12±4\sqrt{22}}{4} when ± is minus. Subtract 4\sqrt{22} from -12.

$$x=\frac{-4\sqrt{22}-12}{4}$$

Divide -12-4\sqrt{22} by 4.

$$x=-\sqrt{22}-3$$

The equation is now solved.

$$x=\sqrt{22}-3 x=-\sqrt{22}-3$$

Steps for Completing the Square

Add x to both sides.

$$2x^{2}+11x-23+x=3$$

Combine 11x and x to get 12x.

$$2x^{2}+12x-23=3$$

Add 23 to both sides.

$$2x^{2}+12x=3+23$$

Add 3 and 23 to get 26.

$$2x^{2}+12x=26$$

Divide both sides by 2.

$$\frac{2x^{2}+12x}{2}=\frac{26}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{12}{2}x=\frac{26}{2}$$

Divide 12 by 2.

$$x^{2}+6x=\frac{26}{2}$$

Divide 26 by 2.

$$x^{2}+6x=13$$

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.

$$x^{2}+6x+3^{2}=13+3^{2}$$

Square 3.

$$x^{2}+6x+9=13+9$$

Add 13 to 9.

$$x^{2}+6x+9=22$$

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}.

$$\left(x+3\right)^{2}=22$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{22}$$

Simplify.

$$x+3=\sqrt{22} x+3=-\sqrt{22}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{22}-3 x=-\sqrt{22}-3$$

Solve for x

$$x=\sqrt{22}-3\approx 1.69041576$$

$$x=-\sqrt{22}-3\approx -7.69041576$$

Steps Using the Quadratic Formula

Add x to both sides.

$$2x^{2}+11x-23+x=3$$

Combine 11x and x to get 12x.

$$2x^{2}+12x-23=3$$

Subtract 3 from both sides.

$$2x^{2}+12x-23-3=0$$

Subtract 3 from -23 to get -26.

$$2x^{2}+12x-26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 12 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 2\left(-26\right)}}{2\times 2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 2\left(-26\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-12±\sqrt{144-8\left(-26\right)}}{2\times 2}$$

Multiply -8 times -26.

$$x=\frac{-12±\sqrt{144+208}}{2\times 2}$$

Add 144 to 208.

$$x=\frac{-12±\sqrt{352}}{2\times 2}$$

Take the square root of 352.

$$x=\frac{-12±4\sqrt{22}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-12±4\sqrt{22}}{4}$$

Now solve the equation x=\frac{-12±4\sqrt{22}}{4} when ± is plus. Add -12 to 4\sqrt{22}.

$$x=\frac{4\sqrt{22}-12}{4}$$

Divide -12+4\sqrt{22} by 4.

$$x=\sqrt{22}-3$$

Now solve the equation x=\frac{-12±4\sqrt{22}}{4} when ± is minus. Subtract 4\sqrt{22} from -12.

$$x=\frac{-4\sqrt{22}-12}{4}$$

Divide -12-4\sqrt{22} by 4.

$$x=-\sqrt{22}-3$$

The equation is now solved.

$$x=\sqrt{22}-3 x=-\sqrt{22}-3$$

Steps for Completing the Square

Add x to both sides.

$$2x^{2}+11x-23+x=3$$

Combine 11x and x to get 12x.

$$2x^{2}+12x-23=3$$

Add 23 to both sides.

$$2x^{2}+12x=3+23$$

Add 3 and 23 to get 26.

$$2x^{2}+12x=26$$

Divide both sides by 2.

$$\frac{2x^{2}+12x}{2}=\frac{26}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{12}{2}x=\frac{26}{2}$$

Divide 12 by 2.

$$x^{2}+6x=\frac{26}{2}$$

Divide 26 by 2.

$$x^{2}+6x=13$$

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.

$$x^{2}+6x+3^{2}=13+3^{2}$$

Square 3.

$$x^{2}+6x+9=13+9$$

Add 13 to 9.

$$x^{2}+6x+9=22$$

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}.

$$\left(x+3\right)^{2}=22$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{22}$$

Simplify.

$$x+3=\sqrt{22} x+3=-\sqrt{22}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{22}-3 x=-\sqrt{22}-3$$