Question

Solve the following equation by factorisation:

$1=\frac{1}{2}{x}^2-\frac{7}{6}x$

Answer 1

Step 1: Simplifying the given equation:

We have, $1=\frac{1}{2}{x}^2-\frac{7}{6}x$

Rearrange the equation in the form of $A{x}^2+Bx+C=0$.

$\begin{array}{rcl}\frac{1}{2}{x}^2-\frac{7}{6}x-1& =& 0\\ \frac{3x^2-7x-6}{6}& =& 0\end{array}$

Multiplying $6$ on both sides.

$\begin{array}{rcl}3x^2-7x-6& =& 0\end{array}$

Step 2: Factorizing the simplified equation:

$\begin{array}{rcl}& \Rightarrow & 3x^2-7x-6=0\\ & \Rightarrow & 3x^2-9x+2x-6=0\\ & \Rightarrow & 3x(x-3)-2(x-3)=0\\ & \Rightarrow & (3x-2)(x-3)=0\end{array}$

Using zero-product property find the value of $x$.

$\begin{array}{rcl}3x-2& =& 0\\ x& =& \frac{2}{3}\end{array}$

Or,

$\begin{array}{rcl}x-3& =& 0\\ x& =& 3\end{array}$

Therefore, the values are $x=\frac{2}{3}$ or, $x=3$.