Factor Factor: 2x^2+5x-3

The given problem is a factoring problem in algebra. The goal here is to express the quadratic trinomial 2x^2+5x-3 as a product of two binomial expressions. This involves finding two numbers that multiply to give the product of the leading coefficient (2) and the constant term (-3), while also adding up to the middle coefficient (5). The solution would typically involve the application of techniques such as the 'ac method' or trial-and-error to find the correct pair of numbers that satisfy these conditions, and then rewriting the quadratic expression as two binomials that multiply to give the original trinomial.

Factor: $2 x^{2} + 5 x - 3$

Answer

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Solution:

Step 1:

To factor a quadratic equation in the form $a x^{2} + b x + c$ , identify two numbers that multiply to $a c = 2 \cdot - 3 = - 6$ and add up to $b = 5$ .

Step 1.1:

Extract the coefficient of $x$ from the term $5 x$ . This gives us $2 x^{2} + 5 (x) - 3$ .

Step 1.2:

Decompose the number $5$ into two numbers, $- 1$ and $6$ , that add up to $5$ . This results in $2 x^{2} + (- 1 + 6) x - 3$ .

Step 1.3:

Use the distributive property to separate the terms. We get $2 x^{2} - x + 6 x - 3$ .

Step 2:

Identify and factor out the greatest common factor (GCF) from each pair of terms.

Step 2.1:

Pair the terms for factoring: $(2 x^{2} - x) + (6 x - 3)$ .

Step 2.2:

Factor out the GCF from each pair: $x (2 x - 1) + 3 (2 x - 1)$ .

Step 3:

Extract the common binomial factor, which is $(2 x - 1)$ , to complete the factorization: $(2 x - 1) (x + 3)$ .

Knowledge Notes:

To factor a quadratic polynomial of the form $a x^{2} + b x + c$ , one common method is to use the factoring by grouping approach. Here are the relevant knowledge points for this approach:

Identifying a and c: The coefficients 'a' and 'c' are the first and last coefficients of the quadratic polynomial, respectively.
Product and Sum: Find two numbers that multiply to $a c$ and add to 'b'. These two numbers are crucial for rewriting the middle term 'bx'.
Rewriting the Middle Term: The middle term 'bx' is split into two terms using the numbers found in the previous step.
Distributive Property: Apply the distributive property to group terms that can be factored by the greatest common factor (GCF).
Factoring by Grouping: Group the terms into pairs and factor out the GCF from each pair.
Common Binomial Factor: After factoring by grouping, if there is a common binomial factor in each group, factor it out to complete the factorization of the polynomial.
Final Factorization: The result is the original polynomial expressed as the product of two binomials.