Solve 2x^2+(43+4)x+2(43)+12=0

$$2 { x }^{ 2 } +(43+4)x+2(43)+12=0$$

Solve for x

$$x=\frac{5\sqrt{57}-47}{4}\approx -2.312706956$$

$$x=\frac{-5\sqrt{57}-47}{4}\approx -21.187293044$$

Steps Using the Quadratic Formula

Add 43 and 4 to get 47.

$$2x^{2}+47x+2\times 43+12=0$$

Multiply 2 and 43 to get 86.

$$2x^{2}+47x+86+12=0$$

Add 86 and 12 to get 98.

$$2x^{2}+47x+98=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 47 for b, and 98 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-47±\sqrt{47^{2}-4\times 2\times 98}}{2\times 2}$$

Square 47.

$$x=\frac{-47±\sqrt{2209-4\times 2\times 98}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-47±\sqrt{2209-8\times 98}}{2\times 2}$$

Multiply -8 times 98.

$$x=\frac{-47±\sqrt{2209-784}}{2\times 2}$$

Add 2209 to -784.

$$x=\frac{-47±\sqrt{1425}}{2\times 2}$$

Take the square root of 1425.

$$x=\frac{-47±5\sqrt{57}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-47±5\sqrt{57}}{4}$$

Now solve the equation x=\frac{-47±5\sqrt{57}}{4} when ± is plus. Add -47 to 5\sqrt{57}.

$$x=\frac{5\sqrt{57}-47}{4}$$

Now solve the equation x=\frac{-47±5\sqrt{57}}{4} when ± is minus. Subtract 5\sqrt{57} from -47.

$$x=\frac{-5\sqrt{57}-47}{4}$$

The equation is now solved.

$$x=\frac{5\sqrt{57}-47}{4} x=\frac{-5\sqrt{57}-47}{4}$$

Steps for Completing the Square

Add 43 and 4 to get 47.

$$2x^{2}+47x+2\times 43+12=0$$

Multiply 2 and 43 to get 86.

$$2x^{2}+47x+86+12=0$$

Add 86 and 12 to get 98.

$$2x^{2}+47x+98=0$$

Subtract 98 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}+47x=-98$$

Divide both sides by 2.

$$\frac{2x^{2}+47x}{2}=-\frac{98}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{47}{2}x=-\frac{98}{2}$$

Divide -98 by 2.

$$x^{2}+\frac{47}{2}x=-49$$

Divide \frac{47}{2}, the coefficient of the x term, by 2 to get \frac{47}{4}. Then add the square of \frac{47}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{47}{2}x+\left(\frac{47}{4}\right)^{2}=-49+\left(\frac{47}{4}\right)^{2}$$

Square \frac{47}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{47}{2}x+\frac{2209}{16}=-49+\frac{2209}{16}$$

Add -49 to \frac{2209}{16}.

$$x^{2}+\frac{47}{2}x+\frac{2209}{16}=\frac{1425}{16}$$

Factor x^{2}+\frac{47}{2}x+\frac{2209}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{47}{4}\right)^{2}=\frac{1425}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{47}{4}\right)^{2}}=\sqrt{\frac{1425}{16}}$$

Simplify.

$$x+\frac{47}{4}=\frac{5\sqrt{57}}{4} x+\frac{47}{4}=-\frac{5\sqrt{57}}{4}$$

Subtract \frac{47}{4} from both sides of the equation.

$$x=\frac{5\sqrt{57}-47}{4} x=\frac{-5\sqrt{57}-47}{4}$$