Show your process.
Solving Quadratic Equations by completing the square.
1. 3x² = 2 -x
2. 6x² - x + 1
(nonsense answers - report)
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Show your process.
Solving Quadratic Equations by completing the square.
1. 3x² = 2 -x
2. 6x² - x + 1
(nonsense answers - report)
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Answer:
1. 3x² = 2 - x:
To solve this equation, we’ll move all the terms to one side‚ so that the equation is in the form ax² + bx + c = 0:
3x² + x - 2 = 0
Now, we’ll complete the square. To do this‚ we take half of the coefficient of x (which is 1/2) and square it (which is 1/4). We then add and subtract this value inside the parentheses:
3x² + x + 1/4 - 2 - 1/4 = 0
Next‚ we group the terms:
(3x² + x + 1/4) - (2 + 1/4) = 0
Simplifying further:
(3x + 1/2)² - (8/4 + 1/4) = 0
(3x + 1/2)² - 9/4 = 0
Now‚ we have a perfect square trinomial, which we can rewrite as:
(3x + 1/2)² = 9/4
Taking the square root of both sides:
3x + 1/2 = ±√(9/4)
3x + 1/2 = ±(3/2)
Now‚ we solve for x:
Case 1: 3x + 1/2 = 3/2
3x = 3/2 - 1/2
3x = 2/2
x = 2/6
x = 1/3
Case 2: 3x + 1/2 = -3/2
3x = -3/2 - 1/2
3x = -4/2
x = -4/6
x = -2/3
Therefore‚ the possible solutions for this equation are x = 1/3 and x = -2/3.
2. 6x² - x + 1:
To solve this equation, we’ll use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For this equation‚ a = 6‚ b = -1‚ and c = 1. Plugging these values into the formula:
x = (-(-1) ± √((-1)² - 4 * 6 * 1)) / (2 * 6)
Simplifying inside the square root:
x = (1 ± √(1 - 24)) / 12
x = (1 ± √(-23)) / 12
Since the square root of a negative number is not a real number‚ the equation has no real solutions.