Question

The difference of the square of a number and $8$ is equal to $7$ times that number. Find the positive solution.

Answer 1

Find the positive solution :

Step 1: Assuming the required value:

Let the required number is $x$.

According to given conditions,

The square of the number $=x^2$.

The difference between the square of a number and $8$$=x^2-8$.

Now,$7$ times the number $=7x$.

so the equation can be written as,

$\begin{array}{rcl}{x}^2-8& =& 7x\end{array}$

Step 2: Solve the equation.

Calculate the factors of the quadratic equation:

$\begin{array}{rcl}{x}^2-8& =& 7x\\ & \Rightarrow & x^2-8-7x=0\\ & \Rightarrow & x^2-7x-8=0\\ & \Rightarrow & x^2-8x+x-8=0\\ & \Rightarrow & x\left(x-8\right)+1\left(x-8\right)=0\\ & \Rightarrow & \left(x+1\right)\left(x-8\right)=0\end{array}$

Step 3: Equate factors to zero.

$$\begin{gathered} \Rightarrow x+1=0 \\ \therefore x=-1 \\ \Rightarrow x-8=0 \\ \therefore x=8 \end{gathered}$$

As $-1$ is the negative solution. The only positive solution is $8$.

Hence, the positive solution is $\mathbf{8}$.