Question

Solve \(x^{3}-9 x=2 x^{2}-18\)

Short answer

The solutions are \(x = -3, 2, 3\).

Step-by-step solution

Move All Terms to One Side

Start by moving all the terms to one side of the equation to set the equation to 0. Subtract both sides by \(2x^2 - 18\): \[x^3 - 9x - 2x^2 + 18 = 0.\]

Rearrange and Combine Like Terms

Combine like terms and arrange in standard form: \[x^3 - 2x^2 - 9x + 18 = 0.\]

Factor by Grouping

Try factoring by grouping. Group terms as follows: \[(x^3 - 2x^2) - (9x - 18) = 0.\] Factor out common factors from each group: \[x^2(x - 2) - 9(x - 2) = 0.\]

Factor Out Common Binomial

Factor out the common binomial factor \(x - 2\): \[(x^2 - 9)(x - 2) = 0.\]

Factor \(x^2 - 9\)

Recognize that \(x^2 - 9\) is a difference of squares: \[(x + 3)(x - 3)(x - 2) = 0.\]

Solve for x

Set each factor equal to zero and solve: \[x + 3 = 0 \implies x = -3\] \[x - 3 = 0 \implies x = 3\] \[x - 2 = 0 \implies x = 2.\]

Key concepts

factoring

Factoring is a powerful tool in algebra, particularly when dealing with polynomial equations. It's the process of breaking down an expression into simpler multiplicative components. In the given problem, we have a cubic equation that was transformed into a simpler form through a series of steps. Initially, we moved all terms to one side, then used factoring by grouping. This technique involves grouping terms with common factors, which makes it easier to break down the polynomial. In our problem, grouping \(x^3 - 9x - 2x^2 + 18 = 0\) into \(x^2(x - 2) - 9(x - 2)\) was a crucial step. Next, we factored out \(x - 2\), a common binomial factor, further simplifying the equation.

difference of squares

The concept of the difference of squares is useful when factoring polynomials. It's based on the identity: \(a^2 - b^2 = (a + b)(a - b)\). In the context of our problem, once we factored the polynomial as \(x^2(x - 2) - 9(x - 2)\), we saw \(x - 2\) as a common factor. This allowed us to rewrite \(x^2 - 9\) as a difference of squares: \(x^2 - 9 = (x + 3)(x - 3)\). Recognizing and applying this identity can significantly simplify solving polynomial equations.

solving cubic equations

To solve cubic equations, like the one given \(x^3 - 9x = 2x^2 - 18\), it's essential to rearrange all terms to create a standard form polynomial: \(ax^3 + bx^2 + cx + d = 0\). This was achieved by moving terms to one side and combining like terms, producing \(x^3 - 2x^2 - 9x + 18 = 0\). Techniques such as factoring by grouping and recognizing common patterns (like the difference of squares) are then applied. Breaking down a cubic equation into linear and quadratic factors simplifies the solving process significantly.

zero product property

The zero product property is pivotal in solving polynomial equations. It states that if the product of two factors is zero, at least one of the factors must be zero. After factoring our original cubic equation into \((x + 3)(x - 3)(x - 2) = 0)\), we used this property to find the values of \(x\) that satisfy the equation. By setting each factor equal to zero: \(x + 3 = 0\), \(x - 3 = 0\), and \(x - 2 = 0\), we solve for the roots \( x = -3 \, \, x = 3 \, \, x = 2\). This illustrates how efficiently the zero product property helps in finding solutions to polynomial equations.