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Magazine Chapter 9: Problem 22
Factor each expression. $$ 0.5 a^{2}-2 a-16 $$
Short Answer
Expert verified
0.5(a - 8)(a + 4)
Step by step solution
01
Identify the equation
The given quadratic expression to factor is \( 0.5a^2 - 2a - 16 \).
02
Multiply all terms by 2
To make the coefficient of \(a^2\) a whole number, multiply each term by 2. This gives \( 2(0.5a^2 - 2a - 16) = 1a^2 - 4a - 32 \), simplifying to \( a^2 - 4a - 32 \).
03
Find factors of the quadratic equation
Now factor \( a^2 - 4a - 32 \). We need two numbers that multiply to -32 and add to -4. These numbers are -8 and 4 because \(-8\cdot4 = -32\) and \(-8 + 4 = -4\).
04
Write the factored form
Use these numbers to write the quadratic equation as a product of two binomials: \( a^2 - 4a - 32 = (a - 8)(a + 4) \).
05
Adjust back to the original expression
Since we initially multiplied the equation by 2, we need to adjust: \( 0.5(a - 8)(a + 4) \). Therefore, the factored form of the original equation is: \( 0.5(a - 8)(a + 4) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
Quadratic equations are a fundamental concept in algebra. They are polynomial equations of the form \[ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants. The highest power of the variable \(x\) in this equation is 2, which is why it's called a quadratic equation. These equations can have two solutions, one solution, or no real solutions.Here are the key steps involved in solving quadratic equations:
- Identify the quadratic equation.
- Factor the quadratic expression, if possible.
- Use the quadratic formula \(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\) when factoring is not straightforward.
- Check your solutions by substituting them back into the original equation.
Factoring
Factoring is an essential technique in algebra for simplifying expressions and solving equations. It involves expressing a polynomial as a product of its factors. In the case of quadratic expressions, factoring typically means writing the polynomial as a product of two binomials.Let's break down the process:
- Identify coefficients: For the quadratic equation, ensure you identify \(a\), \(b\), and \(c\) from \(ax^2 + bx + c\).
- Multiply coefficients: Sometimes, it's useful to multiply all terms by a number to simplify factoring, as seen in our exercise where we multiplied by 2 to rid of the fraction.
- Find factor pairs: For the expression to be factored, find pairs of numbers that multiply to \(ac\) (the product of the leading coefficient and the constant term) and add up to \(b\).
- Write the binomials: Use the factor pairs to break down the middle term into two terms, split the expression, and factor by grouping.
Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is about finding the unknown or putting real-life variables into equations and then solving them. Fundamental to higher mathematics, algebra is used extensively in science, engineering, medicine, and economics.Key topics in algebra include:
- Variables: Symbols that represent unknown values.
- Expressions: Combinations of variables and constants using arithmetic operations.
- Equations: Mathematical statements that assert the equality of two expressions.
- Functions: Relations that uniquely assign output values to input values.
- Inequalities: Statements about the relative size or order of two objects.
