To solve the equation 8x² + 6 = 22x, we first want to rearrange it into a standard quadratic form, which is ax² + bx + c = 0.
We start by moving all terms to one side:
8x² – 22x + 6 = 0
Now, we can use the quadratic formula to find the solutions for x. The quadratic formula is:
x = (-b ± √(b² – 4ac)) / (2a)
In our equation, a = 8, b = -22, and c = 6.
Now we calculate the discriminant (b² – 4ac):
(-22)² – 4 * 8 * 6 = 484 – 192 = 292
Since the discriminant is positive, we have two distinct real solutions. Now let’s calculate:
x = (22 ± √292) / (2 * 8)
Calculating the square root:
√292 ≈ 17.09
Now substituting that back into the formula:
x ≈ (22 ± 17.09) / 16
This gives us two possible solutions:
- x₁ ≈ (22 + 17.09) / 16 ≈ 2.44
- x₂ ≈ (22 – 17.09) / 16 ≈ 0.31
Thus, the solutions to the equation 8x² + 6 = 22x are approximately x ≈ 2.44 and x ≈ 0.31.