Solving a quadratic equation using the quadratic formula.

I'll solve this quadratic equation step by step using the quadratic formula.

The equation is: 6x² - 17x + 12 = 0

From the standard form ax² + bx + c = 0, I can identify:
- a = 6
- b = -17
- c = 12

Using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Let me calculate each part:

Step 1: Calculate -b
-b = -(-17) = 17

Step 2: Calculate b²
b² = (-17)² = 289

Step 3: Calculate 4ac
4ac = 4 × 6 × 12 = 288

Step 4: Calculate b² - 4ac (the discriminant)
b² - 4ac = 289 - 288 = 1

Step 5: Calculate √(b² - 4ac)
√1 = 1

Step 6: Calculate 2a
2a = 2 × 6 = 12

Step 7: Apply the formula
x = (17 ± 1) / 12

This gives us two solutions:

Solution 1: x = (17 + 1) / 12 = 18/12 = 3/2

Solution 2: x = (17 - 1) / 12 = 16/12 = 4/3

Let me verify these answers by plugging them back into the original equation:

For x = 3/2:
6(3/2)² - 17(3/2) + 12
= 6(9/4) - 51/2 + 12
= 54/4 - 51/2 + 12
= 27/2 - 51/2 + 24/2
= (27 - 51 + 24)/2
= 0/2
= 0 ✓

For x = 4/3:
6(4/3)² - 17(4/3) + 12
= 6(16/9) - 68/3 + 12
= 96/9 - 68/3 + 12
= 32/3 - 68/3 + 36/3
= (32 - 68 + 36)/3
= 0/3
= 0 ✓

Both solutions check out!

Final Answer: x = 3/2 or x = 4/3