Question

Find the number of solutions of the equation:
$3{{\text{x}}^3}{\text{ = (}}{{\text{x}}^2}{\text{ + }}\sqrt {18} {\text{x + }}\sqrt {32} {\text{)(}}{{\text{x}}^2}{\text{ - }}\sqrt {18} {\text{x - }}\sqrt {32} {\text{) - 4}}{{\text{x}}^2}$

Answer 1

Hint: Let us use the concept of theory of equations. We will first simplify the given equation and then find the total number of solutions.

Complete step-by-step answer:
Now, first we will simplify the given equation. We will use the property ${\text{(a - b)(a + }}{\text{b) = (}}{{\text{a}}^2}{\text{ - }}{{\text{b}}^2})$ in the right – hand side of the given equation.
Applying the property in the equation, we get
$3{{\text{x}}^3}{\text{ = (}}{{\text{x}}^4}{\text{ - (}}\sqrt {18} {\text{x + }}\sqrt {32} {{\text{)}}^2}{\text{) - 4}}{{\text{x}}^2}$
Solving the above equation,
$3{{\text{x}}^3}{\text{ = (}}{{\text{x}}^4}{\text{ - (18}}{{\text{x}}^2}{\text{ + 32 + 2}}\sqrt {18} \sqrt {32} {\text{)) - 4}}{{\text{x}}^2}$
$3{{\text{x}}^3}{\text{ = }}{{\text{x}}^4}{\text{ - 18}}{{\text{x}}^2}{\text{ - 32 - 2}}\sqrt {18} \sqrt {32} {\text{x - 4}}{{\text{x}}^2}$
${{\text{x}}^4}{\text{ - 3}}{{\text{x}}^3}{\text{ - 22}}{{\text{x}}^2}{\text{ - 2}}\sqrt {18} \sqrt {32} {\text{x - 32 = 0}}$
Further solving the above equation, we get
${{\text{x}}^4}{\text{ - 3}}{{\text{x}}^3}{\text{ - 22}}{{\text{x}}^2}{\text{ - 48x - 32 = 0}}$ ….. (1)
Now, according to the theory of equations to find the number of the equation we have to make the given equation in the simplest form. In other words, we have to place the terms containing x in the descending order of their degree i.e. largest to smallest. The largest degree of the x determines the number of solutions the equation has whether the roots are real or imaginary. So, in the equation (1) we see that the degree of x is in descending order. The largest degree is 4. So, according to the theory of equations, equation (1) has a total of four solutions.
So, the number of solutions of the given equations is 1 as the given equation is a biquadratic equation.

Note: To solve such types of questions in which we have to find the total number of solutions of the given equation we will follow a few steps. If the given equations on both sides are of the same nature (algebraic, trigonometric, logarithmic) we have to simplify the given equation by applying various properties. After it, we will apply the theory of equations to find the number of solutions. If the equations on both sides are of different nature, then we have to draw the graph of each side. After drawing the graph of both sides, wherever both graphs cut each other, such points are the number of solutions. By calculating all the intersection points, a total number of solutions can be found.