Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 138

Chapter 5, Problem 138

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: x(x - 7) = 3.

Verified step by step guidance

Start by expanding the left side of the equation:

x (x - 7) = x \cdot x - x \cdot 7 = x^{2} - 7x

Rewrite the equation with the expanded form:

x^{2} - 7x = 3

Bring all terms to one side to set the equation equal to zero:

x^{2} - 7x - 3 = 0

Identify the coefficients for the quadratic equation in standard form

ax^2 + bx + c = 0

: here,

a = 1

b = -7

, and

c = -3

Use the quadratic formula to solve for

x

x = \(\frac{-b \pm \sqrt{b^2 - 4ac}\)}{2a}

. Substitute the values of

a

b

, and

c

into the formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Expanding and Simplifying Algebraic Expressions

This involves applying the distributive property to multiply terms within parentheses and then combining like terms. For example, expanding x(x - 7) results in x² - 7x, which simplifies the equation for easier manipulation.

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Solving them can involve factoring, completing the square, or using the quadratic formula to find the values of x that satisfy the equation.

Setting Equations to Zero

To solve quadratic equations, it is essential to rewrite the equation so that one side equals zero. This allows the use of factoring or the quadratic formula by isolating all terms on one side, creating a standard form ax² + bx + c = 0.

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Related Practice

Textbook Question

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

log_b (xy)⁵ = (log_b x + log_b y)⁵

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Textbook Question

Exercises 137–139 will help you prepare for the material covered in the next section. Solve for x: a(x - 2) = b(2x + 3)

Textbook Question

Without using a calculator, find the exact value of: [log₃ 81 - log_𝝅 1]/[log_2√2 8 - log 0.001]

Textbook Question

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: (x + 2)/(4x + 3) = 1/x

Textbook Question

If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.

Textbook Question

Without using a calculator, find the exact value of log₄ [log₃ (log₂ 8)].