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To factor the quadratic expression [tex]\( a x^2 + b x + c \)[/tex], we need to follow a structured approach. Here is a detailed step-by-step solution:
### Step 1: Identify the Coefficients
Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] in the quadratic expression:
[tex]\[ ax^2 + bx + c \][/tex]
### Step 2: Verify the Expression
Ensure that the quadratic expression is in the correct standard form of [tex]\( ax^2 + bx + c \)[/tex]. In this case, the expression is [tex]\( a x^2 + b x + c \)[/tex].
### Step 3: Look for Common Factors
First, check if there is any common factor for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. If there is, factor it out. For simplicity, we assume there are no common factors other than 1.
### Step 4: Factor by Decomposition (if possible)
Typically, to factor a quadratic expression of the form [tex]\( a x^2 + b x + c \)[/tex], you need to find two numbers that:
- Multiply to [tex]\( ac \)[/tex] (the product of the coefficient of [tex]\( x^2 \)[/tex] and the constant term)
- Add up to [tex]\( b \)[/tex] (the coefficient of [tex]\( x \)[/tex])
Since a direct factorization into a product of binomials may not always be straightforward without specific values for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], tools like the quadratic formula or completing the square are often used.
### Step 5: Rewrite the Expression (if possible)
Given the general nature of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], we can rewrite the quadratic expression using the quadratic formula to find its roots if necessary:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Using the roots, the expression can be factored as:
[tex]\[ a (x - x_1)(x - x_2) \][/tex]
### Step 6: Final Factored Form
However, since we're given that the expression [tex]\( a x^2 + b x + c \)[/tex] does not factor into a simpler product of binomials under general assumptions, the final factored form of the quadratic expression is essentially the expression itself:
[tex]\[ a x^2 + b x + c \][/tex]
Thus, the quadratic expression [tex]\( a x^2 + b x + c \)[/tex] remains in its expanded form:
[tex]\[ a x^2 + b x + c \][/tex]
In conclusion, the expression [tex]\( a x^2 + b x + c \)[/tex] does not simplify further without specific values for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. It remains in the standard quadratic form.
### Step 1: Identify the Coefficients
Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] in the quadratic expression:
[tex]\[ ax^2 + bx + c \][/tex]
### Step 2: Verify the Expression
Ensure that the quadratic expression is in the correct standard form of [tex]\( ax^2 + bx + c \)[/tex]. In this case, the expression is [tex]\( a x^2 + b x + c \)[/tex].
### Step 3: Look for Common Factors
First, check if there is any common factor for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. If there is, factor it out. For simplicity, we assume there are no common factors other than 1.
### Step 4: Factor by Decomposition (if possible)
Typically, to factor a quadratic expression of the form [tex]\( a x^2 + b x + c \)[/tex], you need to find two numbers that:
- Multiply to [tex]\( ac \)[/tex] (the product of the coefficient of [tex]\( x^2 \)[/tex] and the constant term)
- Add up to [tex]\( b \)[/tex] (the coefficient of [tex]\( x \)[/tex])
Since a direct factorization into a product of binomials may not always be straightforward without specific values for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], tools like the quadratic formula or completing the square are often used.
### Step 5: Rewrite the Expression (if possible)
Given the general nature of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], we can rewrite the quadratic expression using the quadratic formula to find its roots if necessary:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Using the roots, the expression can be factored as:
[tex]\[ a (x - x_1)(x - x_2) \][/tex]
### Step 6: Final Factored Form
However, since we're given that the expression [tex]\( a x^2 + b x + c \)[/tex] does not factor into a simpler product of binomials under general assumptions, the final factored form of the quadratic expression is essentially the expression itself:
[tex]\[ a x^2 + b x + c \][/tex]
Thus, the quadratic expression [tex]\( a x^2 + b x + c \)[/tex] remains in its expanded form:
[tex]\[ a x^2 + b x + c \][/tex]
In conclusion, the expression [tex]\( a x^2 + b x + c \)[/tex] does not simplify further without specific values for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. It remains in the standard quadratic form.
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