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Magazine Chapter 5: Problem 30
Factor each trinomial. See Examples 2 and 3 or Example 11. $$ c^{2}-24 c+44 $$
Short Answer
Expert verified
\((c - 22)(c - 2)\)
Step by step solution
01
Identify the Trinomial
We begin by observing that the expression is a trinomial in the form of \( ax^2 + bx + c \). Here, the trinomial is \( c^2 - 24c + 44 \) with \( a = 1 \), \( b = -24 \), and \( c = 44 \).
02
Determine Factorability using Discriminant
To quickly check if the trinomial can be factored over the integers, compute the discriminant using \( b^2 - 4ac \). Substitute the values: \((-24)^2 - 4(1)(44) = 576 - 176 = 400\). Since the discriminant is a perfect square, the trinomial is factorable over the integers.
03
Use the Quadratic Formula
Since the trinomial can be factored, use the quadratic formula \( c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots. Here, \( c = \frac{-(-24) \pm \sqrt{400}}{2(1)} \). Simplify to \( c = \frac{24 \pm 20}{2} \). This gives roots \( c = 22 \) and \( c = 2 \).
04
Express Trinomial as Product of Binomials
The roots of the trinomial suggest it can be expressed as \((c - 22)(c - 2)\). Distribute to verify: \((c - 22)(c - 2) = c^2 - 2c - 22c + 44 = c^2 - 24c + 44\). This confirms the factorization is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Expressions
Quadratic expressions are a cornerstone in algebra and consist of polynomial expressions of degree two. The general form is given by \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These expressions often describe a parabola when graphed on a coordinate plane.
- The term \( ax^2 \) is the quadratic term because it involves \( x \) squared.
- The term \( bx \) is the linear term, describing the slope.
- The term \( c \) is the constant term, affecting the vertical position of the parabola.
Discriminant
The discriminant is a key tool in determining the nature of solutions of a quadratic equation \( ax^2 + bx + c = 0 \). You find it using the formula \( b^2 - 4ac \). The discriminant helps us understand how many and what kind of solutions we can expect.
- If the discriminant is positive and a perfect square, the quadratic can be factored over the integers. For instance, a discriminant of 400 in our step-by-step solution indicates that two distinct integer solutions are present.
- If it is just positive but not a perfect square, the solutions are irrational.
- If it equals zero, the quadratic has exactly one real solution, or a double root.
- If negative, no real solutions exist; instead, the solutions are complex numbers.
Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations. It allows us to find the roots of any quadratic equation \( ax^2 + bx + c = 0 \) and is given by:\[ c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula derives solutions even when other methods like factoring are not possible. Using the quadratic formula ensures you can determine accurate roots of the expression.
- It is crucial to correctly compute the discriminant (\( b^2 - 4ac \)) as it determines the nature of the roots.
- Applying \( -b \) in the formula involves changing the sign of \( b \), ensuring clarity and reducing error.
- Using \( \pm \) indicates that typically, two solutions are available unless the discriminant equals zero.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into its product of simpler polynomials. For quadratics, this often results in the expression of a trinomial as a product of two binomials.
- In our example, the trinomial \( c^2 - 24c + 44 \) was rearranged into \( (c - 22)(c - 2) \), indicating the roots \( c = 22 \) and \( c = 2 \).
- Successful factorization simplifies solving and graphs the polynomial for analysis.
- Correct factorization can be quickly checked by expanding the binomials back into the trinomial form to ensure accuracy, as shown in the solution when verifying \( (c - 22)(c - 2) = c^2 - 24c + 44 \).
- Proper factoring reveals the roots of the equation, which helps us better understand its properties.
