To find the solutions of the equation 8x² + 6 = 22x, we first need to rearrange and organize the equation into a standard quadratic form, which is ax² + bx + c = 0.
We start with the original equation:
8x² + 6 - 22x = 0
Rearranging this gives us:
8x² - 22x + 6 = 0
Next, we can simplify the equation by dividing through by 2:
4x² - 11x + 3 = 0
Now, we can use the quadratic formula to find the solutions. The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
In our case, a = 4, b = -11, and c = 3. Plugging these values into the formula:
x = (11 ± √((-11)² - 4 * 4 * 3)) / (2 * 4)
This simplifies to:
x = (11 ± √(121 - 48)) / 8
Further simplifying gives:
x = (11 ± √73) / 8
Now we have our two potential solutions:
x₁ = (11 + √73) / 8 x₂ = (11 - √73) / 8
Calculating the approximate numerical values:
- x₁ ≈ 2.89
- x₂ ≈ 0.11
Therefore, the solutions to the equation 8x² + 6 = 22x are:
- x₁ ≈ 2.89
- x₂ ≈ 0.11
These results indicate the x-values at which the original quadratic equation holds true. To verify, you can substitute these values back into the original equation to ensure both sides remain equal.