Solve 6x^2+12x+14=7x^2-5

$$6 x ^ { 2 } + 12 x + 14 = 7 x ^ { 2 } - 5$$

Solve for x

$$x=\sqrt{55}+6\approx 13.416198487$$

$$x=6-\sqrt{55}\approx -1.416198487$$

Steps Using the Quadratic Formula

Subtract 7x^{2} from both sides.

$$6x^{2}+12x+14-7x^{2}=-5$$

Combine 6x^{2} and -7x^{2} to get -x^{2}.

$$-x^{2}+12x+14=-5$$

Add 5 to both sides.

$$-x^{2}+12x+14+5=0$$

Add 14 and 5 to get 19.

$$-x^{2}+12x+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 12 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-1\right)\times 19}}{2\left(-1\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-1\right)\times 19}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-12±\sqrt{144+4\times 19}}{2\left(-1\right)}$$

Multiply 4 times 19.

$$x=\frac{-12±\sqrt{144+76}}{2\left(-1\right)}$$

Add 144 to 76.

$$x=\frac{-12±\sqrt{220}}{2\left(-1\right)}$$

Take the square root of 220.

$$x=\frac{-12±2\sqrt{55}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12±2\sqrt{55}}{-2}$$

Now solve the equation x=\frac{-12±2\sqrt{55}}{-2} when ± is plus. Add -12 to 2\sqrt{55}.

$$x=\frac{2\sqrt{55}-12}{-2}$$

Divide -12+2\sqrt{55} by -2.

$$x=6-\sqrt{55}$$

Now solve the equation x=\frac{-12±2\sqrt{55}}{-2} when ± is minus. Subtract 2\sqrt{55} from -12.

$$x=\frac{-2\sqrt{55}-12}{-2}$$

Divide -12-2\sqrt{55} by -2.

$$x=\sqrt{55}+6$$

The equation is now solved.

$$x=6-\sqrt{55} x=\sqrt{55}+6$$

Steps for Completing the Square

Subtract 7x^{2} from both sides.

$$6x^{2}+12x+14-7x^{2}=-5$$

Combine 6x^{2} and -7x^{2} to get -x^{2}.

$$-x^{2}+12x+14=-5$$

Subtract 14 from both sides.

$$-x^{2}+12x=-5-14$$

Subtract 14 from -5 to get -19.

$$-x^{2}+12x=-19$$

Divide both sides by -1.

$$\frac{-x^{2}+12x}{-1}=-\frac{19}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{12}{-1}x=-\frac{19}{-1}$$

Divide 12 by -1.

$$x^{2}-12x=-\frac{19}{-1}$$

Divide -19 by -1.

$$x^{2}-12x=19$$

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.

$$x^{2}-12x+\left(-6\right)^{2}=19+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=19+36$$

Add 19 to 36.

$$x^{2}-12x+36=55$$

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}.

$$\left(x-6\right)^{2}=55$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{55}$$

Simplify.

$$x-6=\sqrt{55} x-6=-\sqrt{55}$$

Add 6 to both sides of the equation.

$$x=\sqrt{55}+6 x=6-\sqrt{55}$$