Question

Check whether the following equation is a quadratic equation.
${x^3} - 4{x^2} - x + 1 = {(x - 2)^3}$

Answer 1

Hint: Use fundamentals of algebra.

Complete step-by-step answer:

Here we have given the equation ${x^3} - 4{x^2} - x + 1 = {(x - 2)^3}$ . We’ll use the formula ${(a - b)^3} = {a^3} - {b^3} - 3{a^2}b - 3a{b^2}$ in the right-hand side of the equation. It’ll give us:
\[\
  {x^3} - 4{x^2} - x + 1 = {(x - 2)^3} \\
   \Rightarrow {x^3} - 4{x^2} - x + 1 = {x^3} - {2^3} - 3 \times {x^2} \times 2 - 3 \times x \times {2^2}{\text{ [}}{(a - b)^3} = {a^3} - {b^3} - 3{a^2}b - 3a{b^2}{\text{]}} \\
\ \]
Now, we’ll take everything to the right-hand side and solve further
\[\
  {{{x}}^3} - 4{x^2} - x + 1 = {{{x}}^3} - {2^3} - 3 \times {x^2} \times 2 - 3 \times x \times {2^2} \\
   \Rightarrow - 4{x^2} + 6{x^2} - x + 12x + 1 + 8 = 0 \\
   \Rightarrow 2{x^2} + 11x + 9 = 0 \\
\ \]
Since the highest power of the polynomial is 2 hence the given equation is quadratic.

Note: don’t judge the polynomial just after looking at it. Better to solve and come to the conclusion. At first glance the equation might look like it’s not quadratic but cubic.