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Magazine Chapter 1: Problem 75
Factor the expression completely. $$2 x^{2}+5 x+3$$
Short Answer
Expert verified
The expression factors to \((x + 1)(2x + 3)\).
Step by step solution
01
Understand the Problem
We need to factor the quadratic expression \(2x^2 + 5x + 3\). Factoring involves rewriting the expression as a product of two binomials.
02
Identify Coefficients
The quadratic expression is of the form \(ax^2 + bx + c\), where \(a = 2\), \(b = 5\), and \(c = 3\). These coefficients will be used to guide our factoring process.
03
Apply the AC Method
Multiply \(a\) and \(c\) to get \(2 \times 3 = 6\). We need to find two numbers that multiply to \(6\) and add to \(b = 5\). These numbers are \(2\) and \(3\) because \(2 \times 3 = 6\) and \(2 + 3 = 5\).
04
Rewrite the Middle Term
Use the numbers from Step 3 to split the middle term \(5x\) into two terms: \(2x + 3x\). Rewrite the expression as \(2x^2 + 2x + 3x + 3\).
05
Factor by Grouping
Group the terms: \((2x^2 + 2x) + (3x + 3)\). Factor out common factors in each group: \(2x(x + 1) + 3(x + 1)\).
06
Factor Out the Common Binomial
Both terms contain the common factor \(x + 1\). Factor it out: \((x + 1)(2x + 3)\).
07
Verify the Factors
Expand \((x + 1)(2x + 3)\) to ensure it equals the original expression. \((x + 1)(2x + 3) = 2x^2 + 3x + 2x + 3 = 2x^2 + 5x + 3\), which matches the original, confirming our factorization.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations, especially when factoring is complex or challenging. A quadratic equation is usually in the form \( ax^2 + bx + c = 0 \). The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula allows you to find the roots of the quadratic equation, which are the values of \( x \) where the equation equals zero.
Understanding each component is crucial:
Understanding each component is crucial:
- \(a\), \(b\), and \(c\) are coefficients from the quadratic expression \(ax^2 + bx + c\).
- The sign \( \pm \) indicates two possible solutions, as you can add or subtract the square root.
- The term \( b^2 - 4ac \) is called the discriminant, which determines the nature of the roots.
Coefficients in Quadratics
Coefficients in a quadratic equation are the numerical values that multiply the variable terms. In an expression \( ax^2 + bx + c \), each numeral serves a distinct role.
For example, in the expression \(2x^2+5x+3\):
- \(a\): This is the coefficient of \(x^2\), indicating how many \(x^2\) terms there are.
- \(b\): This is the coefficient of \(x\), showing how many \(x\) terms there are.
- \(c\): This is the constant term, the term without \(x\).
For example, in the expression \(2x^2+5x+3\):
- The coefficient \(a = 2\) means each square term is doubled.
- The coefficient \(b = 5\) represents the linear influence of \(x\).
- The constant \(c = 3\) is the standalone number affecting the expression's value.
AC Method
The AC Method is a systematic approach to factor quadratic expressions more easily. It is particularly useful when \(a\), the coefficient of \(x^2\), is not equal to 1. This method divides the middle term into two parts that help factor by grouping.
To apply the AC Method, follow these steps:
To apply the AC Method, follow these steps:
- Multiply \(a\) and \(c\) (the leading and constant coefficients) to find their product.
- Find two numbers that multiply to the product of \(ac\) and sum to \(b\) (the middle coefficient).
- Multiply \(a\) and \(c\): \(2 \times 3 = 6\).
- Find two numbers that multiply to 6 and add up to 5: they are 2 and 3.
- Split the linear term \(5x\) into \(2x + 3x\) to facilitate grouping.
