Solve 1+3x(1-x)=0

$$1+3x(1-x)=0$$

Solve for x

$$x=\frac{\sqrt{21}}{6}+\frac{1}{2}\approx 1.263762616$$

$$x=-\frac{\sqrt{21}}{6}+\frac{1}{2}\approx -0.263762616$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by 1-x.

$$1+3x-3x^{2}=0$$

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.

$$-3x^{2}+3x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 3 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-3\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-3±\sqrt{9+12}}{2\left(-3\right)}$$

Add 9 to 12.

$$x=\frac{-3±\sqrt{21}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-3±\sqrt{21}}{-6}$$

Now solve the equation x=\frac{-3±\sqrt{21}}{-6} when ± is plus. Add -3 to \sqrt{21}.

$$x=\frac{\sqrt{21}-3}{-6}$$

Divide -3+\sqrt{21} by -6.

$$x=-\frac{\sqrt{21}}{6}+\frac{1}{2}$$

Now solve the equation x=\frac{-3±\sqrt{21}}{-6} when ± is minus. Subtract \sqrt{21} from -3.

$$x=\frac{-\sqrt{21}-3}{-6}$$

Divide -3-\sqrt{21} by -6.

$$x=\frac{\sqrt{21}}{6}+\frac{1}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{21}}{6}+\frac{1}{2} x=\frac{\sqrt{21}}{6}+\frac{1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3x by 1-x.

$$1+3x-3x^{2}=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$3x-3x^{2}=-1$$

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.

$$-3x^{2}+3x=-1$$

Divide both sides by -3.

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

Dividing by -3 undoes the multiplication by -3.

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

Divide 3 by -3.

$$x^{2}-x=-\frac{1}{-3}$$

Divide -1 by -3.

$$x^{2}-x=\frac{1}{3}$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{1}{3}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add \frac{1}{3} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{7}{12}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{7}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{7}{12}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{21}}{6} x-\frac{1}{2}=-\frac{\sqrt{21}}{6}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{21}}{6}+\frac{1}{2} x=-\frac{\sqrt{21}}{6}+\frac{1}{2}$$