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Chapter 5: Problem 14

Solve each equation, and check the solutions. $$ (6 x+5)(x+4)=0 $$

Short Answer

Expert verified
-5/6 and -4

Step by step solution

01

Apply the Zero Product Property

The equation is \[(6x + 5)(x + 4) = 0\]. According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, set each factor equal to zero: \[6x + 5 = 0\] and \[x + 4 = 0\].
02

Solve for x in the First Equation

Solve the first equation \[6x + 5 = 0\] for \[x\]: \[6x + 5 = 0\] \[6x = -5\] \[x = -\frac{5}{6}\].
03

Solve for x in the Second Equation

Solve the second equation \[x + 4 = 0\] for \[x\]: \[x + 4 = 0\] \[x = -4\].
04

Verify the Solutions

Substitute \[x = -\frac{5}{6}\] and \[x = -4\] back into the original equation to verify. For \[x = -\frac{5}{6}\]: \[(6(-\frac{5}{6}) + 5)(-\frac{5}{6} + 4) = 0\] \[(0)(-\frac{5}{6} + 4) = 0\] which is true.For \[x = -4\]: \[(6(-4) + 5)(-4 + 4) = 0\] \[(-19)(0) = 0\] which is true. Both solutions are correct.

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Key Concepts

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

Zero Product Property
The zero product property is a fundamental principle when solving quadratic equations. It states that if the product of two terms is zero, then at least one of the terms must be zero. In mathematical terms, if \[(A \cdot B = 0)\], then either \[A = 0\], \[B = 0\], or both. This property allows us to break down complex equations into simpler, solvable linear equations. For example, in the equation \[(6x + 5)(x + 4) = 0\], using the zero product property, we can set up two separate equations \[6x + 5 = 0\] and \[x + 4 = 0\]. By solving these individual equations, we can find the values of \[x\] that satisfy the original equation.
Solving Linear Equations
Once we have applied the zero product property, we need to solve the resulting linear equations. Let's break down the process with our example:
1. Solving \[6x + 5 = 0\]:
  • First, isolate \[x\] by subtracting \[5\] from both sides: \[6x = -5\]
  • Next, divide by \[6\] to solve for \[x\]: \[x = -\frac{5}{6}\]
2. Solving \[x + 4 = 0\]:
  • Simply subtract \[4\] from both sides to get \[x = -4\]
After solving these linear equations, we obtain the potential solutions for \[x\]: \[-\frac{5}{6}\] and \[-4\].
Verification of Solutions
After finding potential solutions, it's crucial to verify them by plugging them back into the original equation. Let's check our solutions for the equation \[(6x + 5)(x + 4) = 0\]:
1. For \[x = -\frac{5}{6}\], substitute back into the equation:
\[(6( - \frac{5}{6}) + 5)( - \frac{5}{6} + 4) = 0\]
\[(0)(\frac{19}{6}) = 0\]
This holds true, confirming that \[x = -\frac{5}{6}\] is a valid solution.
2. For \[x = -4\], substitute back into the equation:
\[(6(-4) + 5)(-4 + 4) = 0\]
\[(-19)(0) = 0\]
This is also true, confirming that \[x = -4\] is a valid solution.
Verification helps ensure our solutions are correct and the original equation is satisfied.

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