Solve 91x^2/3+4x=1

$$\frac { 91 x ^ { 2 } } { 3 } + 4 x = 1$$

Solve for x

$$x=\frac{\sqrt{309}-6}{91}\approx 0.127235119$$

$$x=\frac{-\sqrt{309}-6}{91}\approx -0.259103251$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$91x^{2}+12x=3$$

Subtract 3 from both sides.

$$91x^{2}+12x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 91 for a, 12 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 91\left(-3\right)}}{2\times 91}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 91\left(-3\right)}}{2\times 91}$$

Multiply -4 times 91.

$$x=\frac{-12±\sqrt{144-364\left(-3\right)}}{2\times 91}$$

Multiply -364 times -3.

$$x=\frac{-12±\sqrt{144+1092}}{2\times 91}$$

Add 144 to 1092.

$$x=\frac{-12±\sqrt{1236}}{2\times 91}$$

Take the square root of 1236.

$$x=\frac{-12±2\sqrt{309}}{2\times 91}$$

Multiply 2 times 91.

$$x=\frac{-12±2\sqrt{309}}{182}$$

Now solve the equation x=\frac{-12±2\sqrt{309}}{182} when ± is plus. Add -12 to 2\sqrt{309}.

$$x=\frac{2\sqrt{309}-12}{182}$$

Divide -12+2\sqrt{309} by 182.

$$x=\frac{\sqrt{309}-6}{91}$$

Now solve the equation x=\frac{-12±2\sqrt{309}}{182} when ± is minus. Subtract 2\sqrt{309} from -12.

$$x=\frac{-2\sqrt{309}-12}{182}$$

Divide -12-2\sqrt{309} by 182.

$$x=\frac{-\sqrt{309}-6}{91}$$

The equation is now solved.

$$x=\frac{\sqrt{309}-6}{91} x=\frac{-\sqrt{309}-6}{91}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$91x^{2}+12x=3$$

Divide both sides by 91.

$$\frac{91x^{2}+12x}{91}=\frac{3}{91}$$

Dividing by 91 undoes the multiplication by 91.

$$x^{2}+\frac{12}{91}x=\frac{3}{91}$$

Divide \frac{12}{91}, the coefficient of the x term, by 2 to get \frac{6}{91}. Then add the square of \frac{6}{91} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{12}{91}x+\left(\frac{6}{91}\right)^{2}=\frac{3}{91}+\left(\frac{6}{91}\right)^{2}$$

Square \frac{6}{91} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{12}{91}x+\frac{36}{8281}=\frac{3}{91}+\frac{36}{8281}$$

Add \frac{3}{91} to \frac{36}{8281} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{12}{91}x+\frac{36}{8281}=\frac{309}{8281}$$

Factor x^{2}+\frac{12}{91}x+\frac{36}{8281}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{6}{91}\right)^{2}=\frac{309}{8281}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{6}{91}\right)^{2}}=\sqrt{\frac{309}{8281}}$$

Simplify.

$$x+\frac{6}{91}=\frac{\sqrt{309}}{91} x+\frac{6}{91}=-\frac{\sqrt{309}}{91}$$

Subtract \frac{6}{91} from both sides of the equation.

$$x=\frac{\sqrt{309}-6}{91} x=\frac{-\sqrt{309}-6}{91}$$