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Chapter 6: Problem 304

Solve. $$ 14 y^{2}-77 y=-35 $$

Short Answer

Expert verified
The solutions are \( y = \frac{1}{2} \) and \( y = 5 \).

Step by step solution

01

Rewrite the Equation in Standard Form

Add 35 to both sides to set the equation to zero: \[ 14y^2 - 77y + 35 = 0 \]
02

Factor Out the Greatest Common Factor (GCF)

Identify and factor out the GCF of 14, 77, and 35 from the equation. The GCF is 7: \[ 7(2y^2 - 11y + 5) = 0 \]
03

Factor the Quadratic Expression

Factor the quadratic expression inside the parentheses. We need two numbers that multiply to \(2 \times 5 = 10\) and add up to -11. These numbers are -10 and -1.Rewrite the expression as: \[ 7(2y^2 - 10y - y + 5) = 0 \]Group terms: \[ 7((2y^2 - 10y) + (-y + 5)) = 0 \]Factor each group: \[ 7(2y(y - 5) - 1(y - 5)) = 0 \]Factor out the common term \((y - 5)\): \[ 7(2y - 1)(y - 5) = 0 \]
04

Solve for y

Set each factor equal to zero and solve:1. \(2y - 1 = 0\): \[ 2y = 1 \]\[ y = \frac{1}{2} \]2. \(y - 5 = 0\): \[ y = 5 \]

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Key Concepts

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

Factoring Quadratics
When solving quadratic equations, factoring is one of the key methods used. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \), and factoring involves expressing this as a product of binomials or other factors. To solve by factoring, follow these steps:
  • Rewrite the equation in standard form if needed.
  • Find the greatest common factor (GCF) and factor it out.
  • Factor the resulting simpler quadratic expression.
  • Set each factor equal to zero and solve for the variable.
In our example, the equation was: \(14y^2 - 77y + 35 = 0\). First, we identified and factored out the GCF, and then, computed the factors of the remaining quadratic expression.
Standard Form of Quadratic Equation
The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\). Working with this form is crucial because:
  • It makes it easier to identify coefficients \(a\), \(b\), and \(c\).
  • It鈥檚 a prerequisite for applying factoring, completing the square, or the quadratic formula.
To convert any quadratic equation to standard form, you need to perform algebraic operations to ensure that all terms are on one side of the equation and zero on the other. In the given exercise, the equation \(14y^2 - 77y = -35\) was transformed to standard form by adding 35 to both sides to get: \(14y^2 - 77y + 35 = 0\).
Greatest Common Factor
The concept of the greatest common factor (GCF) helps in simplifying expressions. The GCF of a set of numbers is the largest factor shared by all the numbers.
For example, for numbers 14, 77, and 35, the GCF is 7.
Here's why the GCF is important in factoring:
  • It simplifies the equation, making further factoring easier.
  • It often reveals simpler forms of the quadratic expression.
In the exercise, the GCF was factored out from \(14y^2 - 77y + 35\), giving us: \(7(2y^2 - 11y + 5) = 0\). This resulted in a simpler quadratic expression \(2y^2 - 11y + 5\), which then was factored further to find the solutions.

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Most popular questions from this chapter

In the following exercises, factor each trinomial of the form \(x^{2}+b x y+c y^{2} .\) $$ m^{2}-64 m n-65 n^{2} $$

In the following exercises, factor by grouping. $$ r^{2}-3 r-r+3 $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ a^{2}+25 a+100 $$

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

In the following exercises, factor each expression using any method. $$ 13 z^{2}+39 z-26 $$
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