Solve 6x(x+1)=5-x

$$6 x ( x + 1 ) = 5 - x$$

Solve for x

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

$$x=\frac{1}{2}=0.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x by x+1.

$$6x^{2}+6x=5-x$$

Subtract 5 from both sides.

$$6x^{2}+6x-5=-x$$

Add x to both sides.

$$6x^{2}+6x-5+x=0$$

Combine 6x and x to get 7x.

$$6x^{2}+7x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 7 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 6\left(-5\right)}}{2\times 6}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 6\left(-5\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-7±\sqrt{49-24\left(-5\right)}}{2\times 6}$$

Multiply -24 times -5.

$$x=\frac{-7±\sqrt{49+120}}{2\times 6}$$

Add 49 to 120.

$$x=\frac{-7±\sqrt{169}}{2\times 6}$$

Take the square root of 169.

$$x=\frac{-7±13}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-7±13}{12}$$

Now solve the equation x=\frac{-7±13}{12} when ± is plus. Add -7 to 13.

$$x=\frac{6}{12}$$

Reduce the fraction \frac{6}{12} to lowest terms by extracting and canceling out 6.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{-7±13}{12} when ± is minus. Subtract 13 from -7.

$$x=-\frac{20}{12}$$

Reduce the fraction \frac{-20}{12} to lowest terms by extracting and canceling out 4.

$$x=-\frac{5}{3}$$

The equation is now solved.

$$x=\frac{1}{2} x=-\frac{5}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 6x by x+1.

$$6x^{2}+6x=5-x$$

Add x to both sides.

$$6x^{2}+6x+x=5$$

Combine 6x and x to get 7x.

$$6x^{2}+7x=5$$

Divide both sides by 6.

$$\frac{6x^{2}+7x}{6}=\frac{5}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{7}{6}x=\frac{5}{6}$$

Divide \frac{7}{6}, the coefficient of the x term, by 2 to get \frac{7}{12}. Then add the square of \frac{7}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{6}x+\left(\frac{7}{12}\right)^{2}=\frac{5}{6}+\left(\frac{7}{12}\right)^{2}$$

Square \frac{7}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{5}{6}+\frac{49}{144}$$

Add \frac{5}{6} to \frac{49}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{169}{144}$$

Factor x^{2}+\frac{7}{6}x+\frac{49}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{12}\right)^{2}=\frac{169}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{12}\right)^{2}}=\sqrt{\frac{169}{144}}$$

Simplify.

$$x+\frac{7}{12}=\frac{13}{12} x+\frac{7}{12}=-\frac{13}{12}$$

Subtract \frac{7}{12} from both sides of the equation.

$$x=\frac{1}{2} x=-\frac{5}{3}$$