Solve 6x^2+48x+207=15

$$6 x ^ { 2 } + 48 x + 207 = 15$$

Solve for x (complex solution)

$$x=-4+4i$$

$$x=-4-4i$$

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.

$$6x^{2}+48x+207=15$$

Subtract 15 from both sides of the equation.

$$6x^{2}+48x+207-15=15-15$$

Subtracting 15 from itself leaves 0.

$$6x^{2}+48x+207-15=0$$

Subtract 15 from 207.

$$6x^{2}+48x+192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 48 for b, and 192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 6\times 192}}{2\times 6}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\times 6\times 192}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-48±\sqrt{2304-24\times 192}}{2\times 6}$$

Multiply -24 times 192.

$$x=\frac{-48±\sqrt{2304-4608}}{2\times 6}$$

Add 2304 to -4608.

$$x=\frac{-48±\sqrt{-2304}}{2\times 6}$$

Take the square root of -2304.

$$x=\frac{-48±48i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-48±48i}{12}$$

Now solve the equation x=\frac{-48±48i}{12} when ± is plus. Add -48 to 48i.

$$x=\frac{-48+48i}{12}$$

Divide -48+48i by 12.

$$x=-4+4i$$

Now solve the equation x=\frac{-48±48i}{12} when ± is minus. Subtract 48i from -48.

$$x=\frac{-48-48i}{12}$$

Divide -48-48i by 12.

$$x=-4-4i$$

The equation is now solved.

$$x=-4+4i x=-4-4i$$

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.

$$6x^{2}+48x+207=15$$

Subtract 207 from both sides of the equation.

$$6x^{2}+48x+207-207=15-207$$

Subtracting 207 from itself leaves 0.

$$6x^{2}+48x=15-207$$

Subtract 207 from 15.

$$6x^{2}+48x=-192$$

Divide both sides by 6.

$$\frac{6x^{2}+48x}{6}=-\frac{192}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{48}{6}x=-\frac{192}{6}$$

Divide 48 by 6.

$$x^{2}+8x=-\frac{192}{6}$$

Divide -192 by 6.

$$x^{2}+8x=-32$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-32+4^{2}$$

Square 4.

$$x^{2}+8x+16=-32+16$$

Add -32 to 16.

$$x^{2}+8x+16=-16$$

Factor x^{2}+8x+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}.

$$\left(x+4\right)^{2}=-16$$

Take the square root of both sides of the equation.

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

Simplify.

$$x+4=4i x+4=-4i$$

Subtract 4 from both sides of the equation.

$$x=-4+4i x=-4-4i$$