Question

Solve by completing the square.

$4y^2-6y=4$

Short answer

The solution is$y=2$and$y=-\frac{1}{2}$.

Step-by-step solution

Step 1. Given information.

The given equation is$4y^2-6y=4$.

Step 2. First, isolate the variable and constant terms on one side.

$\begin{array}{rcl}4y^2-6y& =& 4\\ \frac{4y^2}{4}-\frac{6}{4}y& =& \frac{4}{4}\\ y^2-\frac{3}{2}y& =& 1\end{array}$

Step 3. Simplify the equation.

$\begin{array}{rcl}{y}^2-\frac{3}{2}y& =& 1\\ y^2-2·y·\frac{3}{4}& =& 1\\ y^2-2·y·\frac{3}{4}+\frac{9}{16}& =& 1+\frac{9}{16}\cdots \cdots \left(add\frac{9}{16}onbothsides\right)\\ y^2-2·y·\frac{3}{4}+{\left(\frac{3}{4}\right)}^2& =& \frac{25}{16}\\ {\left(y-\frac{3}{4}\right)}^2& =& \frac{25}{16}\\ y-\frac{3}{4}& =& \pm \sqrt{\frac{25}{16}}\\ y-\frac{3}{4}& =& \pm \frac{5}{4}\end{array}$

Step 4. Solve the equation for y.

$\begin{array}{rcl}y-\frac{3}{4}& =& \frac{5}{4}\\ y& =& \frac{5}{4}+\frac{3}{4}\\ y& =& \frac{8}{4}\\ y& =& 2\end{array}$ and$\begin{array}{rcl}y-\frac{3}{4}& =& -\frac{5}{4}\\ y& =& -\frac{5}{4}+\frac{3}{4}\\ y& =& \frac{-5+3}{4}\\ y& =& -\frac{2}{4}\\ y& =& -\frac{1}{2}\end{array}$

Step 5. Simplified answer.

Hence, the required solution is$y=2$and$y=-\frac{1}{2}$.