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Magazine Chapter 1: Problem 29
Find all real solutions of the equation. x^{2}-8 x+12=0
Short Answer
Expert verified
The real solutions are \( x = 6 \) and \( x = 2 \).
Step by step solution
01
Identify the Equation Type
The equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -8 \), and \( c = 12 \).
02
Apply the Quadratic Formula
To find the solutions, we use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Plugging in the values from our equation, we have:\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1} \]
03
Calculate the Discriminant
First, calculate the discriminant, \( b^2 - 4ac \). \[ b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 12 = 64 - 48 = 16 \]
04
Solve for x Using the Quadratic Formula
Since the discriminant is 16, a perfect square, the solutions are real and rational.\[ x = \frac{8 \pm \sqrt{16}}{2} = \frac{8 \pm 4}{2} \]This results in two possible solutions for \( x \):1. \[ x = \frac{8 + 4}{2} = 6 \]2. \[ x = \frac{8 - 4}{2} = 2 \]
05
Verify Solutions
Substitute both solutions back into the original equation to verify:1. For \( x = 6 \), substitute back:\( 6^2 - 8 \times 6 + 12 = 36 - 48 + 12 = 0 \)2. For \( x = 2 \), substitute back:\( 2^2 - 8 \times 2 + 12 = 4 - 16 + 12 = 0 \)Both calculations confirm the solutions are correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Discriminant
In understanding quadratic equations, the discriminant plays a crucial role in determining the nature of the equation's solutions.
The discriminant is the part of the quadratic formula under the square root, expressed as \( b^2 - 4ac \). This component gives insight into whether solutions are real or complex and whether they are distinct or repeated.
In the example provided, the discriminant is \( 16 \), a positive number and a perfect square, indicating two distinct real solutions that are rational numbers.
The discriminant is the part of the quadratic formula under the square root, expressed as \( b^2 - 4ac \). This component gives insight into whether solutions are real or complex and whether they are distinct or repeated.
- If the discriminant is positive, like in our case \( (16) \), the quadratic equation has two distinct real solutions.
- If it is zero, there is exactly one real solution, also known as a repeated or double root.
- If the discriminant is negative, there are no real solutions; instead, the equation has two complex solutions.
In the example provided, the discriminant is \( 16 \), a positive number and a perfect square, indicating two distinct real solutions that are rational numbers.
Quadratic Formula
The quadratic formula is a powerful tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Using this formula allows us to find solutions efficiently without manually factoring the equation.
In our exercise, by substituting the coefficients \( a = 1 \), \( b = -8 \), and \( c = 12 \) into the formula, we proceeded as follows:\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1} \]
In our exercise, by substituting the coefficients \( a = 1 \), \( b = -8 \), and \( c = 12 \) into the formula, we proceeded as follows:\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1} \]
- The 'minus b' operation changes \( -8 \) to a positive 8.
- The discriminant calculation \( b^2 - 4ac \) yields 16, guiding us to the square root part.
Real Solutions
Real solutions in the context of quadratic equations refer to values of \( x \) that satisfy the equation, falling within the set of real numbers. Real solutions are tangible and can often be plotted on a graph.
For the equation \( x^2 - 8x + 12 = 0 \), the discriminant value guides us to determine there are two real solutions.
After computing the quadratic formula, we arrived at these solutions:
For the equation \( x^2 - 8x + 12 = 0 \), the discriminant value guides us to determine there are two real solutions.
After computing the quadratic formula, we arrived at these solutions:
- \( x = \frac{8 + 4}{2} = 6 \)
- \( x = \frac{8 - 4}{2} = 2 \)
Verification of Solutions
Verification of solutions is the final step in solving a quadratic equation to ensure that the solutions found are indeed correct. This is done by substituting the obtained solutions back into the original equation.
For our solutions:
For our solutions:
- Substituting \( x = 6 \) into the equation results in: \( 6^2 - 8 \times 6 + 12 = 36 - 48 + 12 = 0 \)
- Substituting \( x = 2 \) results in: \( 2^2 - 8 \times 2 + 12 = 4 - 16 + 12 = 0 \)
