Solve 72x+12x^2+108=0

$$72 x + 12 x ^ { 2 } + 108 = 0$$

Solve for x

$$x=-3$$

Steps Using the Quadratic Formula

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.

$$12x^{2}+72x+108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 72 for b, and 108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\times 12\times 108}}{2\times 12}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\times 12\times 108}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-72±\sqrt{5184-48\times 108}}{2\times 12}$$

Multiply -48 times 108.

$$x=\frac{-72±\sqrt{5184-5184}}{2\times 12}$$

Add 5184 to -5184.

$$x=\frac{-72±\sqrt{0}}{2\times 12}$$

Take the square root of 0.

$$x=-\frac{72}{2\times 12}$$

Multiply 2 times 12.

$$x=-\frac{72}{24}$$

Divide -72 by 24.

$$x=-3$$

Steps for Completing the Square

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.

$$12x^{2}+72x+108=0$$

Subtract 108 from both sides of the equation.

$$12x^{2}+72x+108-108=-108$$

Subtracting 108 from itself leaves 0.

$$12x^{2}+72x=-108$$

Divide both sides by 12.

$$\frac{12x^{2}+72x}{12}=-\frac{108}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{72}{12}x=-\frac{108}{12}$$

Divide 72 by 12.

$$x^{2}+6x=-\frac{108}{12}$$

Divide -108 by 12.

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

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.

$$x^{2}+6x+3^{2}=-9+3^{2}$$

Square 3.

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

Add -9 to 9.

$$x^{2}+6x+9=0$$

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}.

$$\left(x+3\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+3=0 x+3=0$$

Subtract 3 from both sides of the equation.

$$x=-3 x=-3$$

The equation is now solved. Solutions are the same.

$$x=-3$$