Solve x/2+9=-x^2+4x+3

$$\frac { x } { 2 } + 9 = - x ^ { 2 } + 4 x + 3$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{47}i}{4}\approx 1.75+1.71391365i$$

$$x=\frac{-\sqrt{47}i+7}{4}\approx 1.75-1.71391365i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x+18=-2x^{2}+8x+6$$

Add 2x^{2} to both sides.

$$x+18+2x^{2}=8x+6$$

Subtract 8x from both sides.

$$x+18+2x^{2}-8x=6$$

Combine x and -8x to get -7x.

$$-7x+18+2x^{2}=6$$

Subtract 6 from both sides.

$$-7x+18+2x^{2}-6=0$$

Subtract 6 from 18 to get 12.

$$-7x+12+2x^{2}=0$$

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.

$$2x^{2}-7x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -7 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 2\times 12}}{2\times 2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 2\times 12}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-8\times 12}}{2\times 2}$$

Multiply -8 times 12.

$$x=\frac{-\left(-7\right)±\sqrt{49-96}}{2\times 2}$$

Add 49 to -96.

$$x=\frac{-\left(-7\right)±\sqrt{-47}}{2\times 2}$$

Take the square root of -47.

$$x=\frac{-\left(-7\right)±\sqrt{47}i}{2\times 2}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{47}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{7±\sqrt{47}i}{4}$$

Now solve the equation x=\frac{7±\sqrt{47}i}{4} when ± is plus. Add 7 to i\sqrt{47}.

$$x=\frac{7+\sqrt{47}i}{4}$$

Now solve the equation x=\frac{7±\sqrt{47}i}{4} when ± is minus. Subtract i\sqrt{47} from 7.

$$x=\frac{-\sqrt{47}i+7}{4}$$

The equation is now solved.

$$x=\frac{7+\sqrt{47}i}{4} x=\frac{-\sqrt{47}i+7}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x+18=-2x^{2}+8x+6$$

Add 2x^{2} to both sides.

$$x+18+2x^{2}=8x+6$$

Subtract 8x from both sides.

$$x+18+2x^{2}-8x=6$$

Combine x and -8x to get -7x.

$$-7x+18+2x^{2}=6$$

Subtract 18 from both sides.

$$-7x+2x^{2}=6-18$$

Subtract 18 from 6 to get -12.

$$-7x+2x^{2}=-12$$

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.

$$2x^{2}-7x=-12$$

Divide both sides by 2.

$$\frac{2x^{2}-7x}{2}=-\frac{12}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{7}{2}x=-\frac{12}{2}$$

Divide -12 by 2.

$$x^{2}-\frac{7}{2}x=-6$$

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-6+\left(-\frac{7}{4}\right)^{2}$$

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-6+\frac{49}{16}$$

Add -6 to \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-\frac{47}{16}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{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}.

$$\left(x-\frac{7}{4}\right)^{2}=-\frac{47}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{-\frac{47}{16}}$$

Simplify.

$$x-\frac{7}{4}=\frac{\sqrt{47}i}{4} x-\frac{7}{4}=-\frac{\sqrt{47}i}{4}$$

Add \frac{7}{4} to both sides of the equation.

$$x=\frac{7+\sqrt{47}i}{4} x=\frac{-\sqrt{47}i+7}{4}$$