Solve -14x^2+4x+14=-6x^2

$$- 14 x ^ { 2 } + 4 x + 14 = - 6 x ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{29} + 1}{4} \approx 1.596291202$$

$$x=\frac{1-\sqrt{29}}{4}\approx -1.096291202$$

Steps Using the Quadratic Formula

Add 6x^{2} to both sides.

$$-14x^{2}+4x+14+6x^{2}=0$$

Combine -14x^{2} and 6x^{2} to get -8x^{2}.

$$-8x^{2}+4x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 4 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-8\right)\times 14}}{2\left(-8\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-8\right)\times 14}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-4±\sqrt{16+32\times 14}}{2\left(-8\right)}$$

Multiply 32 times 14.

$$x=\frac{-4±\sqrt{16+448}}{2\left(-8\right)}$$

Add 16 to 448.

$$x=\frac{-4±\sqrt{464}}{2\left(-8\right)}$$

Take the square root of 464.

$$x=\frac{-4±4\sqrt{29}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-4±4\sqrt{29}}{-16}$$

Now solve the equation x=\frac{-4±4\sqrt{29}}{-16} when ± is plus. Add -4 to 4\sqrt{29}.

$$x=\frac{4\sqrt{29}-4}{-16}$$

Divide -4+4\sqrt{29} by -16.

$$x=\frac{1-\sqrt{29}}{4}$$

Now solve the equation x=\frac{-4±4\sqrt{29}}{-16} when ± is minus. Subtract 4\sqrt{29} from -4.

$$x=\frac{-4\sqrt{29}-4}{-16}$$

Divide -4-4\sqrt{29} by -16.

$$x=\frac{\sqrt{29}+1}{4}$$

The equation is now solved.

$$x=\frac{1-\sqrt{29}}{4} x=\frac{\sqrt{29}+1}{4}$$

Steps for Completing the Square

Add 6x^{2} to both sides.

$$-14x^{2}+4x+14+6x^{2}=0$$

Combine -14x^{2} and 6x^{2} to get -8x^{2}.

$$-8x^{2}+4x+14=0$$

Subtract 14 from both sides. Anything subtracted from zero gives its negation.

$$-8x^{2}+4x=-14$$

Divide both sides by -8.

$$\frac{-8x^{2}+4x}{-8}=-\frac{14}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{4}{-8}x=-\frac{14}{-8}$$

Reduce the fraction \frac{4}{-8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{1}{2}x=-\frac{14}{-8}$$

Reduce the fraction \frac{-14}{-8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{2}x=\frac{7}{4}$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{7}{4}+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{7}{4}+\frac{1}{16}$$

Add \frac{7}{4} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{29}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{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}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{29}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{29}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{29}}{4} x-\frac{1}{4}=-\frac{\sqrt{29}}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{\sqrt{29}+1}{4} x=\frac{1-\sqrt{29}}{4}$$