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

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

Solve for x

$$x=2\sqrt{6}-6\approx -1.101020514$$

$$x=-2\sqrt{6}-6\approx -10.898979486$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4.

$$-x^{2}-12x+16=28$$

Subtract 28 from both sides.

$$-x^{2}-12x+16-28=0$$

Subtract 28 from 16 to get -12.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -12 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-12\right)±\sqrt{144+4\left(-12\right)}}{2\left(-1\right)}$$

Multiply 4 times -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-48}}{2\left(-1\right)}$$

Add 144 to -48.

$$x=\frac{-\left(-12\right)±\sqrt{96}}{2\left(-1\right)}$$

Take the square root of 96.

$$x=\frac{-\left(-12\right)±4\sqrt{6}}{2\left(-1\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{6}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{12±4\sqrt{6}}{-2}$$

Now solve the equation x=\frac{12±4\sqrt{6}}{-2} when ± is plus. Add 12 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}+12}{-2}$$

Divide 12+4\sqrt{6} by -2.

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

Now solve the equation x=\frac{12±4\sqrt{6}}{-2} when ± is minus. Subtract 4\sqrt{6} from 12.

$$x=\frac{12-4\sqrt{6}}{-2}$$

Divide 12-4\sqrt{6} by -2.

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

The equation is now solved.

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

Steps for Completing the Square

Multiply both sides of the equation by 4.

$$-x^{2}-12x+16=28$$

Subtract 16 from both sides.

$$-x^{2}-12x=28-16$$

Subtract 16 from 28 to get 12.

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

Divide both sides by -1.

$$\frac{-x^{2}-12x}{-1}=\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{12}{-1}\right)x=\frac{12}{-1}$$

Divide -12 by -1.

$$x^{2}+12x=\frac{12}{-1}$$

Divide 12 by -1.

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

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=-12+6^{2}$$

Square 6.

$$x^{2}+12x+36=-12+36$$

Add -12 to 36.

$$x^{2}+12x+36=24$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=24$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{24}$$

Simplify.

$$x+6=2\sqrt{6} x+6=-2\sqrt{6}$$

Subtract 6 from both sides of the equation.

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