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Chapter 10: Problem 26

For Exercises 7 to \(75,\) factor. $$z^{2}-14 z+49$$

Short Answer

Expert verified
The factored form of the given quadratic equation is \((z - 7)^2\).

Step by step solution

01

Identify the coefficients

The equation is in the form \(z^2 - 14z + 49\), here coefficient \(a = 1\), \(b = -14\) and \(c = 49\).
02

Find two numbers that multiply to \(ac\) and add up to \(b\)

We need to find two numbers that multiply to \(ac = 1*49 = 49\) and add up to \(b = -14\). The numbers are \(-7\) and \(-7\) because \(-7 * -7 = 49\) and \(-7 + -7 = -14\).
03

Write the factored form

The quadratic equation \(z^2 - 14z + 49\) can be factored as \((z - 7)(z - 7)\) or \((z - 7)^2\).

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Key Concepts

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

Quadratic Equation
A quadratic equation is an algebraic expression that takes the general form \(ax^2 + bx + c = 0\). It is called "quadratic" because the highest degree of an exponent in the expression is two. Quadratics are found in various real-world scenarios, such as calculating projectile motion or determining areas. In the context of your textbook problem, the quadratic equation is \(z^2 - 14z + 49\). The primary goal when faced with a quadratic equation is often to find its roots, which involve putting it into a factored form or using the quadratic formula.
Coefficient Identification
Identifying the coefficients in a quadratic equation is a crucial step. In the equation \(z^2 - 14z + 49\), each component plays a specific role:
  • \(a = 1\): This is the coefficient of \(z^2\).
  • \(b = -14\): This is the coefficient of \(z\).
  • \(c = 49\): This is the constant term.
Identifying these numbers accurately allows you to use them in further calculations, such as factoring or applying the quadratic formula. The value of \(a\) being 1 simplifies the calculation process since it doesn鈥檛 complicate the factors needed for the solution.
Factored Form
The factored form of a quadratic equation simplifies it to where it can be written as the product of two binomials. For the quadratic \(z^2 - 14z + 49\), the factored form is \((z - 7)(z - 7)\), which simplifies to \((z - 7)^2\). This means that \(z = 7\) is a root with multiplicity two, indicating there is a repeated root. To convert to factored form:
  • Find two numbers that multiply to \(ac\) (in this case, 49) and add up to \(b\) (in this case, -14).
  • The numbers are both -7, simplifying the equation to its factored form.
The factored form reveals the solutions of the quadratic equation, providing insight into the nature of the roots.
Algebraic Expressions
An algebraic expression involves numbers, variables, and operational symbols. Quadratics, being a type of algebraic expression, follow specific forms and rules that guide their manipulation.In the exercise, \(z^2 - 14z + 49\) is broken down by operations that maintain its equality's balance. The beauty of algebra is using these expressions to model and solve real-life problems. Key skills include manipulating terms, transforming expressions into simpler forms, and understanding their graphical representations.Understanding how to handle these expressions is vital for successfully factoring them, as it leads to efficient problem-solving methods in mathematics.

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