Answer :
Answer:
Analyzing the quadratic expression \( ax^2 + bx + c \) by the manufacturer typically involves examining its coefficients \( a \), \( b \), and \( c \) to understand properties such as the vertex, roots, and axis of symmetry of the corresponding quadratic function. They may also assess the discriminant (\( b^2 - 4ac \)) to determine the nature of the roots (real, complex, or repeated) and utilize this information for various purposes such as optimization, modeling, or solving real-world problems. Is there anything specific you would like to know about the analysis of the quadratic expression?
Step-by-step explanation:
Could you please specify what aspect of the analysis you're interested in? Are you looking for information on how to analyze the roots, vertex, or any other specific property of the quadratic expression \( ax^2 + bx + c \)?
Answer:
This section explains how to factor expressions of the form ax [ 2 ] + bx + c, where a, b, and c are integers.
First, factor out all constants which evenly divide all three terms. If a is negative, factor out -1. This will leave an expression of the form d (ax [ 2 ] + bx + c), where a, b, c, and d are integers, and a > 0. We can now turn to factoring the inside expression.
Here is how to factor an expression ax [ 2 ] + bx + c, where a > 0:
Write out all the pairs of numbers that, when multiplied, produce a.
Write out all the pairs of numbers that, when multiplied, produce c.
Pick one of the a pairs -- (a1, a2) -- and one of the c pairs -- (c1, c2).
If c > 0: Compute a1c1 + a2c2. If | a1c1 + a2c2| = b, then the factored form of the quadratic is
(a1x + c2)(a2x + c1) if b > 0.
(a1x - c2)(a2x - c1) if b < 0.
If a1c1 + a2c2≠b, compute a1c2 + a2c1. If a1c2 + a2c1 = b, then the factored form of the quadratic is (a1x + c1)(a2x + c2) or (a1x + c1)(a2x + c2). If a1c2 + a2c1≠b, pick another set of pairs.
If c < 0: Compute a1c1 -a2c2. If | a1c1 - a2c2| = b, then the factored form of the quadratic is:
(a1x - c2)(a2x + c1) where a1c1 > a2c2 if b > 0 and a1c1 < a2c2 if b < 0.