Question

How do you solve \[{x^2} - 5 = 73\] ?

Answer 1

Hint: In this question, a polynomial equation is given to us, 2 is the highest exponent in the given equation, so the degree of the equation is 2, and hence it is a quadratic equation and has exactly two solutions. Solutions of an equation are defined as the values of the x for which the given function has a value zero or when we plot this function on the graph, we see that the solutions of this function are the points on which the y-coordinate is zero, thus they are simply the x-intercepts.

Complete step-by-step solution:
We are given –
$
  {x^2} - 5 = 73 \\
   \Rightarrow {x^2} = 78 \\
   \Rightarrow x = \pm \sqrt {78} \\
   \Rightarrow x = \pm 8.832 \\
 $
Hence the factors of the equation \[{x^2} - 5 = 73\] are $x - 8.832 = 0$ and $x + 8.832 = 0$.
Here we can also know that $x = 8.832, -8.832$

Note: We know that $a{x^2} + bx + c = 0$ is the standard form of a quadratic polynomial. The methods that are usually used to find the solutions/root/factors of a quadratic equation are factoring or completing the square method or the quadratic formula. But when we compare the given equation with the standard form, we see that the value of b is equal to zero that’s why we simply bring c to the other side of the equal sign and divide both sides by a, then we get the roots by square rooting both sides of the equation. An arithmetic identity can also be used to solve the above question, the identity states that the difference of the square of one number a and the square of another number b is equal to the product of the difference of a and b and the sum of a and b, that is, ${a^2} - {b^2} = (a - b)(a + b)$.