Question

If we have an equation as ${a^2} + \dfrac{9}{{{a^2}}} = 31$ , What is the value of $a - \dfrac{3}{a}$ ?

Answer 1

Hint: To solve this question we should first try to evaluate the given algebraic expression.
Algebraic equation: An algebraic equation is a mathematical sentence, when two algebraic expressions are related with an equality sign.
In this equation, try to simplify it, then we try to apply algebraic identity, then make it the same as in format as we desire our answer then calculate the answer.
Here we will use, ${(a - b)^2} = {a^2} + {b^2} - 2ab$

Complete step-by-step solution:
As given in the question,
${a^2} + \dfrac{9}{{{a^2}}} = 31$
We can rewrite it as,
${(a)^2} + {\left( {\dfrac{3}{a}} \right)^2} = 31$
We can subtract 6 on both side,
${(a)^2} + {\left( {\dfrac{3}{a}} \right)^2} - 6 = 31 - 6$
We can rewrite it as,
${(a)^2} + {\left( {\dfrac{3}{a}} \right)^2} - 2 \times a \times \dfrac{3}{a} = 25$
As we know ${(a - b)^2} = {a^2} + {b^2} - 2ab$
So, ${\left( {a - \dfrac{3}{a}} \right)^2} = 25$
Taking square root both the side,
$\sqrt {{{\left( {a - \dfrac{3}{a}} \right)}^2}} = \sqrt {25} $
Now we open under root as,
 $a - \dfrac{3}{a} = \pm 5$
So, the value of $a - \dfrac{3}{a}$ is $ \pm 5$ .

Note: Types of Algebraic Equations
Algebraic equations are of various types. We should have knowledge of all the algebraic equations for quick solving such problems. A few of the equations in algebra are:
$\bullet$ Polynomial Equations: All the polynomial equations are a part of algebraic equations like the linear equations. To recall, a polynomial equation is an equation consisting of variables, exponents and coefficients.
$\bullet$ Quadratic Equations: A quadratic equation is a polynomial equation of degree 2 in one variable of type \[f\left( x \right) = a{x^2} + bx + c\]
$\bullet$ Cubic Equations: The cubic polynomials are polynomials with degree 3. All the cubic polynomials are also algebraic equations.
Rational polynomial Equations:
$\bullet$ Trigonometric Equations: All the trigonometric equations are all considered as algebraic functions. For a trigonometry equation, the expression includes the trigonometric functions of a variable.