Problem 73 - Algebra 2

Question

Solve each equation. $$ 125 x^{3}-1=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0.2\) and \(x = -0.1 ± 0.6i\).

Step 1: Factor the equation

Recognize that this is a difference of cubes, which can be factored using the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). In this case, \(a = 5x\) and \(b = 1\), so the factored expression becomes \((5x - 1)((5x)^2 + (5x)(1) + (1)^2) = 0\). Simplify this to \((5x - 1)(25x^2 + 5x + 1) = 0\).

Step 2: Set each factor equal to zero and solve

Set each factored term equal to zero and solve for \(x\). Firstly, consider \((5x - 1) = 0\). Adding 1 to both sides and then dividing by 5 gives \(x = 0.2\). Secondly, for fist part of the equation with imaginary roots, \(25x^2 + 5x + 1 = 0\), to find the solutions, we need to use the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)] / (2a)\). Here, \(a = 25, b = 5, c = 1\), by substituting these values, \(x = [-5 ± sqrt((5^2) - 4*(25)*(1))] / (2 * 25)\) simplifies to \(x = [-5 ± sqrt(25 - 100)] / 50\). This gives \(x = -0.1 ± 0.6i\).

Step 3: Checking the result

Finally check if the found answers, \(x = 0.2\) and \(x = -0.1 ± 0.6i\), are indeed the solutions by putting these values back in \(125x^3 - 1 = 0\). If they return true, the process found correct solutions.

Key Concepts

Factoring Cubic Equations

Factoring cubic equations is an essential skill in algebra that helps simplify and solve polynomial equations of degree three. The key to factoring these equations is recognizing common patterns, such as the difference of cubes. A cubic equation can sometimes be rewritten into simpler polynomial expressions that, when multiplied together, will provide the original equation. This can often involve moving terms around and using known formulas.

For example, the exercise provided involves a cubic expression in the form of a difference of cubes: \(125x^3 - 1 = 0\). Recognizing these as perfect cubes, we can apply the difference of cubes formula:

• Given the general form \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), it simplifies these expressions.
• Identifying \(a = 5x\) and \(b = 1\) gives us two factors: \((5x - 1)(25x^2 + 5x + 1) = 0\).

This step is crucial since it reduces a complex problem into solvable parts. By breaking down cubic expressions, finding simpler, solvable equations becomes much more manageable.

Quadratic Formula

The quadratic formula is a versatile and powerful technique for finding solutions to quadratic equations of the form \(ax^2 + bx + c = 0\). When factoring isn't possible, or the equation isn't easily simplified, the quadratic formula provides exact results. It is written as:

• \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\)

Using the quadratic formula involves identifying the coefficients \(a\), \(b\), and \(c\) from the equation and substituting them into the formula.

In our example, the factor \((25x^2 + 5x + 1)\) was solved using this formula to find the roots of the quadratic equation. By substituting \(a = 25\), \(b = 5\), and \(c = 1\), the discriminant step \(b^2 - 4ac\) is calculated as \(25 - 100 = -75\). Consequently, it results in a negative number, indicating the presence of complex solutions.

Complex Numbers

Complex numbers arise when equations have solutions that cannot be expressed as real numbers. This happens often when the discriminant \(b^2 - 4ac\) is negative in the quadratic formula. Complex numbers have the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) represents the imaginary unit, defined as \(\sqrt{-1}\).

In the original step by step solution, the quadratic equation \(25x^2 + 5x + 1 = 0\) gives complex roots because the discriminant is \(-75\). By rewriting the square root of a negative number, we introduce the imaginary unit \(i\):

• The roots become \(-0.1 \pm 0.6i\).
• Each solution is a combination of a real part \(-0.1\) and an imaginary part \(\pm 0.6i\).

Understanding complex solutions is crucial in mathematics and engineering, as they describe real-world phenomena, like electrical currents, waves, and more.

Difference of Cubes

The difference of cubes is a specific factoring technique used to simplify expressions like \(a^3 - b^3\). Recognizing perfect cubes can make the factorization process straightforward and reduce complex equations into easier parts. The difference of cubes formula is:

• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

This formula requires identifying the cube roots \(a\) and \(b\) from the terms in the equation.

For the equation \(125x^3 - 1 = 0\), this is

• a case of the difference of cubes where \(a = 5x\) and \(b = 1\).
• This allows quick rewriting as \((5x - 1)(25x^2 + 5x + 1) = 0\).

Mastering this formula not only makes solving cubic equations easier but also applies to broader algebraic and calculus contexts.