Solve 5z(2z+3)=2z+3

$$5 z ( 2 z + 3 ) = 2 z + 3$$

Solve for z

$$z = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$z=\frac{1}{5}=0.2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5z by 2z+3.

$$10z^{2}+15z=2z+3$$

Subtract 2z from both sides.

$$10z^{2}+15z-2z=3$$

Combine 15z and -2z to get 13z.

$$10z^{2}+13z=3$$

Subtract 3 from both sides.

$$10z^{2}+13z-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 13 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-13±\sqrt{13^{2}-4\times 10\left(-3\right)}}{2\times 10}$$

Square 13.

$$z=\frac{-13±\sqrt{169-4\times 10\left(-3\right)}}{2\times 10}$$

Multiply -4 times 10.

$$z=\frac{-13±\sqrt{169-40\left(-3\right)}}{2\times 10}$$

Multiply -40 times -3.

$$z=\frac{-13±\sqrt{169+120}}{2\times 10}$$

Add 169 to 120.

$$z=\frac{-13±\sqrt{289}}{2\times 10}$$

Take the square root of 289.

$$z=\frac{-13±17}{2\times 10}$$

Multiply 2 times 10.

$$z=\frac{-13±17}{20}$$

Now solve the equation z=\frac{-13±17}{20} when ± is plus. Add -13 to 17.

$$z=\frac{4}{20}$$

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

$$z=\frac{1}{5}$$

Now solve the equation z=\frac{-13±17}{20} when ± is minus. Subtract 17 from -13.

$$z=-\frac{30}{20}$$

Reduce the fraction \frac{-30}{20} to lowest terms by extracting and canceling out 10.

$$z=-\frac{3}{2}$$

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 5z by 2z+3.

$$10z^{2}+15z=2z+3$$

Subtract 2z from both sides.

$$10z^{2}+15z-2z=3$$

Combine 15z and -2z to get 13z.

$$10z^{2}+13z=3$$

Divide both sides by 10.

$$\frac{10z^{2}+13z}{10}=\frac{3}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$z^{2}+\frac{13}{10}z=\frac{3}{10}$$

Divide \frac{13}{10}, the coefficient of the x term, by 2 to get \frac{13}{20}. Then add the square of \frac{13}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}+\frac{13}{10}z+\left(\frac{13}{20}\right)^{2}=\frac{3}{10}+\left(\frac{13}{20}\right)^{2}$$

Square \frac{13}{20} by squaring both the numerator and the denominator of the fraction.

$$z^{2}+\frac{13}{10}z+\frac{169}{400}=\frac{3}{10}+\frac{169}{400}$$

Add \frac{3}{10} to \frac{169}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}+\frac{13}{10}z+\frac{169}{400}=\frac{289}{400}$$

Factor z^{2}+\frac{13}{10}z+\frac{169}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z+\frac{13}{20}\right)^{2}=\frac{289}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+\frac{13}{20}\right)^{2}}=\sqrt{\frac{289}{400}}$$

Simplify.

$$z+\frac{13}{20}=\frac{17}{20} z+\frac{13}{20}=-\frac{17}{20}$$

Subtract \frac{13}{20} from both sides of the equation.

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