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Chapter 5: Problem 95

Factor. $$t^{2}-9 t+14$$

Short Answer

Expert verified
The factorized form of the given quadratic expression is \( (t-7)(t-2) \).

Step by step solution

01

Identify the coefficients

The given expression is a quadratic expression in the form of \( ax^{2} + bx + c \), where \( a = 1, b = -9, c = 14 \).
02

Find two numbers

Find two numbers that multiply to 14 (the value of c) and add up to -9 (the value of b). These numbers are -7 and -2, because -7 multiplied by -2 equals 14, and -7 added to -2 equals -9.
03

Rewrite the original quadratic

Substitute these numbers into the original quadratic to rewrite it as a factored expression. So, \( t^{2} - 9t + 14 \) becomes \( (t-7)(t-2) \).

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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 mathematical statements that involve variables raised to the second power. They are typically written in the form \( ax^2 + bx + c \). Here, the variable, usually denoted as \( x \) or any other letter, represents an unknown value, while \( a \), \( b \), and \( c \) are constants called coefficients.Quadratic expressions are important in both pure and applied mathematics, as they can model various scenarios such as projectile motion or the shape of a parabolic mirror. Key characteristics of quadratic expressions include:
  • The highest power of the variable is 2, making it a polynomial of degree 2.
  • They can graph as parabolas in a coordinate plane.
  • The terms must be arranged in descending order of power of the variable.
Coefficients
Coefficients are the numbers in front of the variables in an algebraic expression. In our quadratic expression \( t^2 - 9t + 14 \), the coefficients are easy to identify:- \( a = 1 \), which is the coefficient of \( t^2 \).- \( b = -9 \), which is the coefficient of \( t \).- \( c = 14 \), the constant term, or coefficient of \( t^0 \).Coefficients determine the direction, width, and position of the parabola when you graph a quadratic expression:
  • When \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
  • The value of \( b \) affects the symmetry and direction of the parabola along the x-axis.
  • The coefficient \( c \) is the y-intercept of the parabola.
Understanding these coefficients is crucial for efficiently transforming quadratic expressions into factored forms that reveal crucial points like roots/zeros.
Factored Expressions
Factored expressions are another way of writing algebraic expressions, and they make equations simpler to analyze and solve. For the quadratic \( t^2 - 9t + 14 \), the expression is factored into \( (t-7)(t-2) \).Factoring involves breaking down the expression into simpler products that, when multiplied, return to the original expression. It's a helpful process because:
  • It simplifies solving the equation \( ax^2 + bx + c = 0 \) by using the zero-product property, which states that if a product is zero, one or more of the factors must be zero.
  • Factored forms expose the roots of the quadratic, which are \( t = 7 \) and \( t = 2 \) in this example, highlighting the values where the expression equals zero.
Using factoring, we can quickly determine the solutions to quadratic equations or understand their graph characteristics without extensive computation.

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Most popular questions from this chapter

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