Question

Find the real solutions, if any, of each equation. $$ 5 a^{3}-45 a=-2 a^{2}+18 $$

Short answer

The real solutions are \(a = -\frac{2}{5}, a = -3, \text{and} a = 3\).

Step-by-step solution

Move all terms to one side

First, move all terms to one side of the equation to set it equal to zero:\[5a^3 - 45a + 2a^2 - 18 = 0\]

Combine like terms

Combine the like terms on the left side of the equation:\[5a^3 + 2a^2 - 45a - 18 = 0\]

Factor by grouping

Next, factor by grouping. First, group the terms in pairs:\[(5a^3 + 2a^2) + (-45a - 18) = 0\]Now factor out the greatest common factor from each pair:\[a^2(5a + 2) - 9(5a + 2) = 0\]Notice that \(5a + 2\) is a common factor.

Extract common factors

Factor out \(5a + 2\) from the expression:\[(5a + 2)(a^2 - 9) = 0\]

Factor the quadratic expression

Factor the quadratic expression \(a^2 - 9\) as a difference of squares:\[(5a + 2)(a + 3)(a - 3) = 0\]

Solve for the roots

Set each factor equal to zero and solve for \(a\):1. For \(5a + 2 = 0\):\[5a + 2 = 0\]\[5a = -2\]\[a = -\frac{2}{5}\]2. For \(a + 3 = 0\):\[a + 3 = 0\]\[a = -3\]3. For \(a - 3 = 0\):\[a - 3 = 0\]\[a = 3\]

Key concepts

Factoring by Grouping

Factoring by grouping is a useful method for simplifying polynomial equations, especially when the terms can be arranged into pairs with a common factor. To apply this method, begin by grouping terms that share a common variable or number.
If we have the polynomial expression \(5a^3 + 2a^2 - 45a - 18\), a logical step is to group terms such as \(5a^3 + 2a^2\) and \(-45a - 18\). Ideally, each group should be simplified to reveal a common factor.
In our example, when we factor out \(a^2\) from the first group and \(-9\) from the second group, we're left with:\(a^2(5a + 2) - 9(5a + 2)\).
Both groups now share \(5a + 2\) as a factor, which allows for further simplification.

Difference of Squares

The difference of squares technique is instrumental in factoring certain types of polynomials. This specific formula is based on the identity \[a^2 - b^2 = (a + b)(a - b)\].
In the exercise, we encounter the expression \(a^2 - 9\). Recognizing that \(9\) is a perfect square (\(3^2\)), we can rewrite \(a^2 - 9\) as \(a^2 - 3^2\).
Using the difference of squares formula, this expression becomes \((a + 3)(a - 3)\).
This step simplifies the polynomial further and allows us to find all its roots.

Solving Polynomial Equations

Solving polynomial equations involves finding the values of the variable that make the equation true. In the given exercise, after factoring the polynomial, we are left with \((5a + 2)(a + 3)(a - 3) = 0\).
The next step is to apply the zero-product property, which states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). This is because zero multiplied by any number is still zero.
To find the solutions, we set each factor to zero and solve for \(a\). The three factors are:

• \(5a + 2 = 0\)
• \(a + 3 = 0\)
• \(a - 3 = 0\)

Solving these individually gives us the roots of the polynomial.

Roots of Polynomial

The roots of a polynomial are the values of the variable that satisfy the polynomial equation, making it equal to zero. In our exercise, the roots correspond to the points where \((5a + 2)(a + 3)(a - 3)=0\).
From the zero-product property, setting \(5a + 2 = 0\) yields \(a = -\frac{2}{5}\); setting \(a + 3 = 0\) gives \(a = -3\); and setting \(a - 3 = 0\) gives \(a = 3\).
These values are the solutions of our original equation. Each root represents an intersection of the graph of the polynomial with the x-axis, showing where the polynomial equals zero.
Understanding these roots is crucial in many aspects of mathematics, from graphing functions to solving real-world problems.