Solve 4xleft(3x+1right)=5

$$4x \left( 3x+1 \right) = 5$$

Solve for x

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

$$x=-\frac{5}{6}\approx -0.833333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by 3x+1.

$$12x^{2}+4x=5$$

Subtract 5 from both sides.

$$12x^{2}+4x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 4 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 12\left(-5\right)}}{2\times 12}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 12\left(-5\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-4±\sqrt{16-48\left(-5\right)}}{2\times 12}$$

Multiply -48 times -5.

$$x=\frac{-4±\sqrt{16+240}}{2\times 12}$$

Add 16 to 240.

$$x=\frac{-4±\sqrt{256}}{2\times 12}$$

Take the square root of 256.

$$x=\frac{-4±16}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-4±16}{24}$$

Now solve the equation x=\frac{-4±16}{24} when ± is plus. Add -4 to 16.

$$x=\frac{12}{24}$$

Reduce the fraction \frac{12}{24} to lowest terms by extracting and canceling out 12.

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

Now solve the equation x=\frac{-4±16}{24} when ± is minus. Subtract 16 from -4.

$$x=-\frac{20}{24}$$

Reduce the fraction \frac{-20}{24} to lowest terms by extracting and canceling out 4.

$$x=-\frac{5}{6}$$

The equation is now solved.

$$x=\frac{1}{2} x=-\frac{5}{6}$$

Steps for Completing the Square

Use the distributive property to multiply 4x by 3x+1.

$$12x^{2}+4x=5$$

Divide both sides by 12.

$$\frac{12x^{2}+4x}{12}=\frac{5}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{4}{12}x=\frac{5}{12}$$

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

$$x^{2}+\frac{1}{3}x=\frac{5}{12}$$

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=\frac{5}{12}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{5}{12}+\frac{1}{36}$$

Add \frac{5}{12} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{4}{9}$$

Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=\frac{4}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{\frac{4}{9}}$$

Simplify.

$$x+\frac{1}{6}=\frac{2}{3} x+\frac{1}{6}=-\frac{2}{3}$$

Subtract \frac{1}{6} from both sides of the equation.

$$x=\frac{1}{2} x=-\frac{5}{6}$$