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Magazine Chapter 10: Problem 83
Solve each equation. See Sections 2.3 and 6.6 $$ x^{2}-8 x=-12 $$
Short Answer
Expert verified
The solutions are \( x = 6 \) and \( x = 2 \).
Step by step solution
01
Write the equation in standard quadratic form
The given equation is \( x^2 - 8x = -12 \). To write it in standard quadratic form, move all terms to one side of the equation. Add 12 to both sides of the equation to obtain: \( x^2 - 8x + 12 = 0 \).
02
Factor the quadratic equation
To solve the equation \( x^2 - 8x + 12 = 0 \), we look for two numbers whose product is 12 and whose sum is -8. These numbers are -6 and -2. Thus, the equation factors as \( (x - 6)(x - 2) = 0 \).
03
Solve the factored equation
Set each factor equal to zero and solve for \( x \): \( x - 6 = 0 \) or \( x - 2 = 0 \). This gives the solutions \( x = 6 \) and \( x = 2 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring
When it comes to quadratic equations, factoring is a fundamental technique used to find the solutions or roots of the equation. It's like breaking down a complex problem into simpler parts. To factor a quadratic equation, you need to express it as a product of two binomials. A quadratic equation generally looks like this:
- \( ax^2 + bx + c = 0 \)
- Identify coefficients: Look at the quadratic equation and identify the coefficients \( a, b, \) and \( c\) . These play a crucial role in factoring.
- Finding numbers: The goal is to find two numbers that multiply to \( ac \) (the coefficient of the \( x^2 \) term times the constant term) and add up to \( b \) (the coefficient of the \( x \) term). In our exercise, we needed numbers that multiply to 12 and add up to -8, which are -6 and -2.
- Set up binomials: Using these numbers, you set up the factors in the form \((x - m)(x - n)\), where \( m \) and \( n \) are the numbers you found. For our equation, it was \((x - 6)(x - 2) = 0\).
Standard Quadratic Form
Before you can factor a quadratic equation or use other methods to solve it, you need to express it in standard quadratic form. This form of the equation looks like:
- \( ax^2 + bx + c = 0 \)
- Combine terms: Move all terms to one side of the equation so that zero is on the other side. This often involves adding or subtracting terms.
- Simplify: Make sure that like terms are combined, such that you end up with a neat arrangement of terms involving \( x^2 \), \( x \), and constant values.
- Check coefficients: Confirm that \( a, b, \) and \( c \) are readily visible in the equation, where \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant.
Solution Methods
Once a quadratic equation is factored or prepared in its standard form, there are several solution methods you can use to find its roots or solutions. Let's dive into the primary ones:
- Factoring: This is often the simplest method if the quadratic can be easily factored. Once in the factored form, set each factor equal to zero and solve for the variable.
- Example: For \((x - 6)(x - 2) = 0\), solving gives \(x = 6\) and \(x = 2\).
- Quadratic formula: If the quadratic equation is not easily factorable, use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula is universal and works for any quadratic equation, providing both real and complex solutions.
- Completing the square: This technique involves rearranging the equation into a perfect square trinomial, which then makes it easier to solve for \( x \) by taking the square root of both sides.
- Graphing: Solve the equation by graphing the quadratic function and identifying the points where it crosses the x-axis (the roots).
