Solve 6x+84*2x^2=0

$$6 x + 84 \cdot 2 x ^ { 2 } = 0$$

Solve for x

$$x=-\frac{1}{28}\approx -0.035714286$$

$$x=0$$

Steps Using Factoring

Factor out x.

$$x\left(6+2\times 84x\right)=0$$

To find equation solutions, solve x=0 and 6+168x=0.

$$x=0 x=-\frac{1}{28}$$

Steps Using the Quadratic Formula

Multiply 84 and 2 to get 168.

$$6x+168x^{2}=0$$

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.

$$168x^{2}+6x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 168 for a, 6 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}}}{2\times 168}$$

Take the square root of 6^{2}.

$$x=\frac{-6±6}{2\times 168}$$

Multiply 2 times 168.

$$x=\frac{-6±6}{336}$$

Now solve the equation x=\frac{-6±6}{336} when ± is plus. Add -6 to 6.

$$x=\frac{0}{336}$$

Divide 0 by 336.

$$x=0$$

Now solve the equation x=\frac{-6±6}{336} when ± is minus. Subtract 6 from -6.

$$x=-\frac{12}{336}$$

Reduce the fraction \frac{-12}{336} to lowest terms by extracting and canceling out 12.

$$x=-\frac{1}{28}$$

The equation is now solved.

$$x=0 x=-\frac{1}{28}$$

Steps for Completing the Square

Multiply 84 and 2 to get 168.

$$6x+168x^{2}=0$$

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.

$$168x^{2}+6x=0$$

Divide both sides by 168.

$$\frac{168x^{2}+6x}{168}=\frac{0}{168}$$

Dividing by 168 undoes the multiplication by 168.

$$x^{2}+\frac{6}{168}x=\frac{0}{168}$$

Reduce the fraction \frac{6}{168} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{1}{28}x=\frac{0}{168}$$

Divide 0 by 168.

$$x^{2}+\frac{1}{28}x=0$$

Divide \frac{1}{28}, the coefficient of the x term, by 2 to get \frac{1}{56}. Then add the square of \frac{1}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{28}x+\left(\frac{1}{56}\right)^{2}=\left(\frac{1}{56}\right)^{2}$$

Square \frac{1}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{28}x+\frac{1}{3136}=\frac{1}{3136}$$

Factor x^{2}+\frac{1}{28}x+\frac{1}{3136}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{56}\right)^{2}=\frac{1}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{56}\right)^{2}}=\sqrt{\frac{1}{3136}}$$

Simplify.

$$x+\frac{1}{56}=\frac{1}{56} x+\frac{1}{56}=-\frac{1}{56}$$

Subtract \frac{1}{56} from both sides of the equation.

$$x=0 x=-\frac{1}{28}$$