Question

How do you solve it ${x^2} - x - 72 = 0$ ?

Answer 1

Hint: Firstly, the given equation is quadratic. The solution of the quadratic equation will be the roots of the equation. The roots of the equation can be found out by using the roots for a quadratic equation formula. We get two values from the formula both of which are the roots of the equation.

Formula used: The roots of a quadratic equation( $a{x^2} + bx + c = 0$ ) can be found by the formula,
$x = \left[ {\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}} \right]$

Complete step-by-step solution:
The given expression is, ${x^2} - x - 72 = 0$
Now use the formula for finding the roots of the above quadratic expression.
The formula is $x = \left[ {\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}} \right]$
Here, $a = 1;b = ( - 1);c = ( - 72)$
Since the value of
${b^2} - 4ac = {( - 1)^2} - 4( - 72) = 289$
Which is $ > 0$. This means that our roots are going to be distinct and real.
On substituting these values in the formula, we get,
$ \Rightarrow x = \left[ {\dfrac{{ - ( - 1) \pm \sqrt {{{( - 1)}^2} - 4 \times 1 \times ( - 72)} }}{{2 \times 1}}} \right]$
First, solve the operations in the square root.
$ \Rightarrow x = \left[ {\dfrac{{ - ( - 1) \pm \sqrt {1 + 288} }}{{2 \times 1}}} \right]$
$ \Rightarrow x = \left[ {\dfrac{{ - ( - 1) \pm \sqrt {289} }}{{2 \times 1}}} \right]$
Write the values of the square root.
$ \Rightarrow x = \left[ {\dfrac{{1 \pm 17}}{2}} \right]$
Simplify further.
$ \Rightarrow x = \left[ {\dfrac{1}{2} \pm \dfrac{{17}}{2}} \right]$
Open the $ \pm $ sign to get two values of $x$
$ \Rightarrow x = \left[ {\dfrac{1}{2} + \dfrac{{17}}{2}} \right];x = \left[ {\dfrac{1}{2} - \dfrac{{17}}{2}} \right]$
Evaluate them
$ \Rightarrow x = \left[ {\dfrac{{18}}{2}} \right];x = \left[ {\dfrac{{ - 16}}{2}} \right]$
$ \Rightarrow x = 9;x = - 8$

Hence the roots of the expression, ${x^2} - x - 72 = 0$ are $x = 9, - 8$.

Note: A polynomial is a mathematical expression which contains one or more variables in sum or subtraction format with different powers. Whenever there is a polynomial that is to be solved, the solution contains the roots of the expression. The number of roots is decided by the degree of the polynomial.