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Chapter 8: Problem 38

Use factoring and the zero product property to solve. $$6 h^{2}+11 h-7=0$$

Short Answer

Expert verified
The solutions are \( h = \frac{1}{2} \) and \( h = -\frac{7}{3} \).

Step by step solution

01

Write down the equation

Start with the given quadratic equation: \(6h^2 + 11h - 7 = 0\)
02

Factor the quadratic equation

To factor the quadratic equation of the form \(ax^2 + bx + c\), look for two numbers that multiply to \(a \cdot c = 6 \cdot (-7) = -42\) and add to \(b = 11\).The numbers that satisfy these conditions are 14 and -3. So rewrite the middle term (11h) using these numbers:\(6h^2 + 14h - 3h - 7 = 0\)
03

Group the terms and factor by grouping

Group the terms to factor by grouping: \((6h^2 + 14h) + (-3h - 7) = 0\)Factor out the greatest common factor from each group:\(2h(3h + 7) - 1(3h + 7) = 0\)
04

Factor out the common binomial factor

Factor out the common binomial factor \((3h + 7)\):\((2h - 1)(3h + 7) = 0\)
05

Set each factor to zero and solve

Apply the zero product property, which states if \(ab = 0\), then either \(a = 0\) or \(b = 0\):\(2h - 1 = 0\) or \(3h + 7 = 0\)Solve each equation separately:For \(2h - 1 = 0\):\(2h = 1\)\(h = \frac{1}{2}\)For \(3h + 7 = 0\):\(3h = -7\)\(h = -\frac{7}{3}\)

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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 polynomial equation of the form:
zero product property
The zero product property is a crucial concept in solving quadratic equations. It states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if you have an equation like:

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