Solve (2t+3)^2=3(2t+3)

$$( 2 t + 3 ) ^ { 2 } = 3 ( 2 t + 3 )$$

Solve for t

$$t=0$$

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

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2t+3\right)^{2}.

$$4t^{2}+12t+9=3\left(2t+3\right)$$

Use the distributive property to multiply 3 by 2t+3.

$$4t^{2}+12t+9=6t+9$$

Subtract 6t from both sides.

$$4t^{2}+12t+9-6t=9$$

Combine 12t and -6t to get 6t.

$$4t^{2}+6t+9=9$$

Subtract 9 from both sides.

$$4t^{2}+6t+9-9=0$$

Subtract 9 from 9 to get 0.

$$4t^{2}+6t=0$$

Factor out t.

$$t\left(4t+6\right)=0$$

To find equation solutions, solve t=0 and 4t+6=0.

$$t=0 t=-\frac{3}{2}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2t+3\right)^{2}.

$$4t^{2}+12t+9=3\left(2t+3\right)$$

Use the distributive property to multiply 3 by 2t+3.

$$4t^{2}+12t+9=6t+9$$

Subtract 6t from both sides.

$$4t^{2}+12t+9-6t=9$$

Combine 12t and -6t to get 6t.

$$4t^{2}+6t+9=9$$

Subtract 9 from both sides.

$$4t^{2}+6t+9-9=0$$

Subtract 9 from 9 to get 0.

$$4t^{2}+6t=0$$

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

$$t=\frac{-6±\sqrt{6^{2}}}{2\times 4}$$

Take the square root of 6^{2}.

$$t=\frac{-6±6}{2\times 4}$$

Multiply 2 times 4.

$$t=\frac{-6±6}{8}$$

Now solve the equation t=\frac{-6±6}{8} when ± is plus. Add -6 to 6.

$$t=\frac{0}{8}$$

Divide 0 by 8.

$$t=0$$

Now solve the equation t=\frac{-6±6}{8} when ± is minus. Subtract 6 from -6.

$$t=-\frac{12}{8}$$

Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

$$t=0 t=-\frac{3}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2t+3\right)^{2}.

$$4t^{2}+12t+9=3\left(2t+3\right)$$

Use the distributive property to multiply 3 by 2t+3.

$$4t^{2}+12t+9=6t+9$$

Subtract 6t from both sides.

$$4t^{2}+12t+9-6t=9$$

Combine 12t and -6t to get 6t.

$$4t^{2}+6t+9=9$$

Subtract 9 from both sides.

$$4t^{2}+6t=9-9$$

Subtract 9 from 9 to get 0.

$$4t^{2}+6t=0$$

Divide both sides by 4.

$$\frac{4t^{2}+6t}{4}=\frac{0}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$t^{2}+\frac{6}{4}t=\frac{0}{4}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$t^{2}+\frac{3}{2}t=\frac{0}{4}$$

Divide 0 by 4.

$$t^{2}+\frac{3}{2}t=0$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{3}{2}t+\left(\frac{3}{4}\right)^{2}=\left(\frac{3}{4}\right)^{2}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{3}{2}t+\frac{9}{16}=\frac{9}{16}$$

Factor t^{2}+\frac{3}{2}t+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{3}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{3}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$t+\frac{3}{4}=\frac{3}{4} t+\frac{3}{4}=-\frac{3}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

$$t=0 t=-\frac{3}{2}$$