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Magazine Chapter 1: Problem 49
Solve the equation by factoring. $$3 x^{2}+5 x=2$$
Short Answer
Expert verified
The solutions are \( x = \frac{1}{3} \) and \( x = -2 \).
Step by step solution
01
Rearrange the Equation
First, rearrange the equation so that one side equals zero. Start with the original equation:\[ 3x^2 + 5x = 2 \]Subtract 2 from both sides of the equation:\[ 3x^2 + 5x - 2 = 0 \]
02
Identify a, b, and c in the Quadratic
The equation \( 3x^2 + 5x - 2 = 0 \) is in the standard quadratic form \( ax^2 + bx + c = 0 \), where:- \( a = 3 \)- \( b = 5 \)- \( c = -2 \)
03
Factor the Quadratic
We need to factor the quadratic expression \( 3x^2 + 5x - 2 \). We will find two numbers that multiply to \( a \cdot c = 3 \times (-2) = -6 \) and add to \( b = 5 \).The numbers that meet these criteria are 6 and -1, because \( 6 \times (-1) = -6 \) and \( 6 + (-1) = 5 \).
04
Write the Expression with Two Middle Terms
Use the numbers found in Step 3 to rewrite the middle term (5x) so:\[ 3x^2 + 6x - x - 2 = 0 \]
05
Factor by Grouping
First, group the terms:\[ (3x^2 + 6x) + (-x - 2) = 0 \]Factor out the greatest common factor from each group:\[ 3x(x + 2) - 1(x + 2) = 0 \]
06
Factor the Expression Further
Notice now we have a common binomial factor \((x + 2)\):\[ (3x - 1)(x + 2) = 0 \]
07
Solve for x
Set each factor equal to zero: 1) \( 3x - 1 = 0 \)2) \( x + 2 = 0 \)Solve the first equation:\[ 3x = 1 \]\[ x = \frac{1}{3} \]Solve the second equation:\[ x = -2 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Quadratics
Factoring quadratics is a method used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). This technique involves expressing the quadratic equation as a product of two binomials. The goal is to transform the equation into a form that is easier to solve by setting each binomial equal to zero.To factor a quadratic equation:
- Ensure the equation is in standard form by moving all terms to one side so that the equation equals zero.
- Identify coefficients \( a \), \( b \), and \( c \) from the equation \( ax^2 + bx + c = 0 \).
- Find two numbers that multiply to \( a \times c \) and add up to \( b \).
- Rewrite the middle term using the two numbers found, effectively splitting it into two separate components.
- Proceed to factor the equation further, often using other techniques like grouping, to find the solution.
Quadratic Formula
The quadratic formula is a universal solver for any quadratic equation. It is particularly useful when factoring is not straightforward. The formula, \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]allows us to find the roots, or solutions, of a quadratic equation.Using the quadratic formula involves:
- Ensuring the equation is arranged in the standard quadratic form \( ax^2 + bx + c = 0 \).
- Substituting the values of \( a \), \( b \), and \( c \) into the formula.
- Calculating the discriminant \( b^2 - 4ac \) as it dictates the nature of the roots (real and distinct, real and equal, or complex).
- Simplifying the remaining expression to find the possible values of \( x \).
Factoring by Grouping
Factoring by grouping is a technique often used when a quadratic equation can be written in a format that makes it possible to group terms to reveal common factors. It鈥檚 an indispensable tool, especially for quadratics that don鈥檛 fit simple factorization paths.Here's how you factor by grouping:
- With the equation in standard form, split the middle term into two terms that add up to the original middle coefficient and whose coefficients multiply to \( a \times c \).
- Group the four terms into two pairs, each capable of simplifying with a common factor.
- Factor out these common factors, which should leave you with a common binomial factor between the grouped terms.
- The final factored expression will be the product of a binomial and a remaining term.
