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Magazine Chapter 8: Problem 76
a. \((x+2)(x-4)=16\) b. \((x+2)(x-4)=-16\)
Short Answer
Expert verified
(a) Solutions: \( x=6 \) and \( x=-4 \). (b) No real solutions.
Step by step solution
01
Expand the Expression
Start by expanding the expression \( (x+2)(x-4) \). Use the distributive property to multiply each term, giving \( x^2 - 4x + 2x - 8 \). Simplify it to \( x^2 - 2x - 8 \).
02
Set the Equation to Zero
For part (a), equate the expanded expression to 16, resulting in \( x^2 - 2x - 8 = 16 \). Move all terms to one side to get \( x^2 - 2x - 24 = 0 \).
03
Solve the Quadratic Equation (Part a)
Use factoring to solve the quadratic equation \( x^2 - 2x - 24 = 0 \). Factor the equation into \( (x-6)(x+4)=0 \). Solve for \( x \), yielding \( x=6 \) and \( x=-4 \). These are the solutions for part (a).
04
Set the Equation for Part (b)
For part (b), equate the expanded expression to -16, resulting in \( x^2 - 2x - 8 = -16 \). Move all terms to one side to get \( x^2 - 2x + 8 = 0 \).
05
Solve the Quadratic Equation (Part b)
Try to factor the equation \( x^2 - 2x + 8 = 0 \), but find that it cannot be factored easily. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a=1 \), \( b=-2 \), \( c=8 \).Calculate the discriminant: \( b^2 - 4ac = 4 - 32 = -28 \). Since the discriminant is negative, there are no real solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Property
The distributive property is a fundamental concept in mathematics that helps in simplifying expressions. It involves two key operations: multiplication and addition (or subtraction). The property can be expressed as:
- For any numbers or expressions: \( a(b + c) = ab + ac \)
- This means you multiply each term inside the bracket by the term outside the bracket.
- Multiply \(x\) by both \(x\) and \(-4\), resulting in \(x^2 - 4x\).
- Next, multiply \(2\) by both \(x\) and \(-4\), resulting in \(2x - 8\).
- Finally, combine these results, yielding \(x^2 - 4x + 2x - 8\).
- Simplify this expression to \(x^2 - 2x - 8\).
Factoring Quadratics
Factoring quadratics is a technique used to solve quadratic equations, particularly when it's possible to express the equation as a product of binomials. For a quadratic equation in the form \( ax^2 + bx + c = 0 \), we search for two numbers that:
- Multiply to \(ac\), the product of the coefficient of \(x^2\) (\(a\)) and the constant term (\(c\)).
- Add up to \(b\), the coefficient of \(x\).
- They multiply to \(-24\) (\(-6 \times 4\)) and add up to \(-2\).
- Thus, the equation can be expressed as \((x-6)(x+4) = 0\).
- Setting each factor equal to zero gives \(x-6 = 0\) or \(x+4 = 0\), leading to solutions \(x = 6\) and \(x = -4\).
Quadratic Formula
The quadratic formula is a powerful method for solving any quadratic equation, even when factoring is not feasible. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). The quadratic formula is written as:
- \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- This formula provides solutions for any quadratic equation by substituting the values of \(a\), \(b\), and \(c\).
- It calculates the roots by assessing the expression inside the square root, called the discriminant \(b^2 - 4ac\).
- If it's positive, there are two distinct real solutions.
- If it's zero, there is exactly one real solution.
- If it's negative, no real solutions exist (though there are complex solutions).
- Here, \(a = 1\), \(b = -2\), and \(c = 8\).
- The discriminant \(b^2 - 4ac = -28\) is negative, indicating there are no real solutions.
