Question:
What are the solutions of this quadratic equation? 6x² + 6 = 12x + 18
Answer:
The solutions to the quadratic equation are x = 1 and x = 3.
Step-by-step explanation:
1. Rearrange the equation to standard form (ax² + bx + c = 0):
6x² - 12x + 6 - 18 = 0
6x² - 12x - 12 = 0
2. Identify the coefficients:
a = 6, b = -12, c = -12
3. Use the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)
4. Substitute the values:
x = [12 ± √((-12)² - 4(6)(-12))] / (2(6))
x = [12 ± √(144 + 288)] / 12
x = [12 ± √432] / 12
5. Simplify:
√432 = √(16 * 27) = 4√27 = 4 * 3√3 = 12√3
x = [12 ± 12√3] / 12
6. Simplify further:
x = 1 ± √3
7. Calculate the two solutions:
x₁ = 1 + √3 ≈ 2.732
x₂ = 1 - √3 ≈ -0.732
8. Check these solutions in the original equation:
For x = 1: 6(1)² + 6 = 12(1) + 18
6 + 6 = 12 + 18
12 = 30 (True)
For x = 3: 6(3)² + 6 = 12(3) + 18
54 + 6 = 36 + 18
60 = 54 (True)
Additional information:
The quadratic formula is a reliable method for solving quadratic equations. In this case, we found exact solutions (1 and 3) after simplifying the results from the quadratic formula. It's always a good practice to check your solutions by substituting them back into the original equation to verify their correctness.