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Magazine Chapter 6: Problem 93
Solve each equation. $$ 3 a^{3}+4 a^{2}+a=0 $$
Short Answer
Expert verified
The solutions are \(a = 0\), \(a = -\frac{1}{3}\), and \(a = -1\).
Step by step solution
01
Identify Common Factor
Look at the equation \(3a^3 + 4a^2 + a = 0\) and notice that each term has a common factor of \(a\). Factoring \(a\) from the equation gives \(a(3a^2 + 4a + 1) = 0\).
02
Apply Zero-Product Property
According to the zero-product property, if a product of factors is zero, then at least one of the factors must be zero. This gives us two equations to solve: \(a = 0\) and \(3a^2 + 4a + 1 = 0\). The solution to the first equation is simply \(a = 0\).
03
Solve the Quadratic Equation
Solve the quadratic equation \(3a^2 + 4a + 1 = 0\) using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 3\), \(b = 4\), \(c = 1\).
04
Calculate Discriminant
Calculate the discriminant \(b^2 - 4ac\). Here, \(b^2 - 4ac = 4^2 - 4 \times 3 \times 1 = 16 - 12 = 4\). Since the discriminant is positive, there are two real solutions.
05
Apply Quadratic Formula
Substitute \(a = 3\), \(b = 4\), and \(c = 1\) into the quadratic formula: \(a = \frac{-4 \pm \sqrt{4}}{2 \times 3}\). Simplifying, \(a = \frac{-4 \pm 2}{6}\). This results in two solutions: \(a = \frac{-4 + 2}{6} = -\frac{1}{3}\) and \(a = \frac{-4 - 2}{6} = -1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring
Factoring is a vital concept when solving polynomial equations. It involves breaking down a polynomial into simpler terms, called factors, which when multiplied together give back the original polynomial.
In the exercise, the equation \(3a^3 + 4a^2 + a = 0\) was factored by identifying a common factor of \(a\) in all terms.
In the exercise, the equation \(3a^3 + 4a^2 + a = 0\) was factored by identifying a common factor of \(a\) in all terms.
- The original equation is expressed as a product of this common factor: \(a(3a^2 + 4a + 1) = 0\).
- This process simplifies the equation and often reveals solutions directly, especially when combined with other mathematical properties like the zero-product property.
Zero-Product Property
The zero-product property is a fundamental principle used after factoring a polynomial equation. It states that if a product of multiple factors is zero, then at least one of the factors must be zero.
This property was applied to the factored equation \(a(3a^2 + 4a + 1) = 0\) in the exercise.
This property was applied to the factored equation \(a(3a^2 + 4a + 1) = 0\) in the exercise.
- From this, two separate equations are derived: \(a = 0\) and \(3a^2 + 4a + 1 = 0\).
- The solution \(a = 0\) was found immediately, demonstrating the efficacy of this property in revealing solutions.
Quadratic Formula
The quadratic formula is a standard tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides a systematic way to find solutions (roots) to these equations.The formula is given by:\[a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In the exercise, it was used to solve \(3a^2 + 4a + 1 = 0\).
- By substituting \(a = 3\), \(b = 4\), and \(c = 1\) into the formula, the possible values of \(a\) were calculated.
- This results in two solutions based on the sign option (\(+\) or \(-\)) in the formula.
Discriminant
The discriminant is a part of the quadratic formula and plays a critical role in determining the nature of the roots of a quadratic equation. It is the expression under the square root in the quadratic formula:\[b^2 - 4ac\]In the provided exercise:
- The discriminant \(4^2 - 4 \times 3 \times 1 = 16 - 12 = 4\) was calculated.
- A positive discriminant indicates that there are two distinct real solutions for the equation.
- If the discriminant is positive, expect two different real roots.
- If it is zero, the equation has exactly one real root.
- A negative discriminant means the equation has two complex roots.
