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Chapter 6: Problem 113

In the following exercises, factor using the 'ac' method. $$ 4 k^{2}-16 k+15 $$

Short Answer

Expert verified
The factors are (2k - 5)(2k - 3).

Step by step solution

01

Identify a, b, and c

The quadratic equation is in the form of ax^2+bx+c. Here, compare it with the provided equation, 4k^2 - 16k + 15. So, a = 4, b = -16, and c = 15.
02

Calculate ac

Multiply a and c: 4 * 15 = 60.
03

Find two numbers that multiply to ac and add to b

Find two numbers whose product is 60 and sum is -16. These numbers are -10 and -6 since (-10) * (-6) = 60 and (-10) + (-6) = -16.
04

Rewrite the middle term

Rewrite the equation 4k^2 - 16k + 15 by splitting the middle term using -10 and -6:4k^2 - 10k - 6k + 15.
05

Group terms

Group the terms in pairs:(4k^2 - 10k) + (-6k + 15).
06

Factor each group

Factor out the greatest common factor (GCF) from each group of terms:2k(2k - 5) - 3(2k - 5).
07

Factor by grouping

Since both terms contain the common binomial factor (2k - 5), factor this out:(2k - 5)(2k - 3).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations
A quadratic equation is a type of polynomial equation of the second degree. This means it has the general form:
  • ax^2 + bx + c = 0
where a, b, and c are constants, and x represents an unknown variable. Quadratic equations always include the variable raised to the power of two (hence the 'quadratic' name). They can have two roots and are commonly solved by, factorization, the quadratic formula, or completing the square.
This problem involves the equation:
  • 4k^2 -16k +15
To solve quadratic equations by factorization, especially when the coefficient of x^2 (or k^2 in this case) is more than 1, we often use the 'ac' method as detailed.
Factorization
Factorization involves breaking down polynomials into simpler 'factors' that, when multiplied together, give the original polynomial. It simplifies solving quadratic equations by converting them into a product of binomials that equal zero.
To factorize the equation 4k^2 - 16k + 15, we use the 'ac' method as follows:
  • Identify coefficients a, b, and c in the equation.
  • Multiply a and c to get a product ac.
  • Find two numbers that multiply to give ac and sum to give b.
  • Rewrite the middle term (the 'bx' term) using these two numbers.
  • Group terms and factor each group.
  • Finally, factor the common binomial factor from the grouped terms.
This systematic approach, especially the grouping step, makes it easier to solve even complex quadratic equations.
Polynomials
Polynomials are expressions consisting of variables and coefficients, constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The degree of a polynomial is the highest power of the variable in its expression. For example, in 4k^2 - 16k + 15, the highest degree is 2, making it a quadratic polynomial.
  • Monomials have one term (e.g., 3x).
  • Binomials have two terms (e.g., x + 1).
  • Trinomials have three terms.
Our goal in factorizing polynomials, such as the example given, is to convert them into a product of binomials or monomials, which is simpler to solve. The 'ac' method works effectively because it utilizes the properties of both multiplication and addition to break down more complex polynomial expressions.

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Most popular questions from this chapter

What is the GCF of \(y^{4}, y^{5},\) and \(y^{10} ?\) Write a general rule that tells you how to find the GCF of \(y^{a}, y^{b},\) and \(y^{c}\)

In the following exercises, factor each expression using any method. $$ (x+2)^{2}-25(x+2)-54 $$

In the following exercises, factor completely using trial and error. $$ -5 y^{3}-30 y^{2}+35 y $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x y+c y^{2} .\) $$ r^{2}+3 r s-28 s^{2} $$

In the following exercises, factor completely using trial and error. $$ -30 q^{3}-140 q^{2}-80 q $$
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