The first step is to rearrange the equation to get it in standard quadratic form, which is ax^2 + bx + c = 0. To do this, we need to move all the terms to one side of the equation:...
6x^2 + 7 = -20
6x^2 + 7 + 20 = 0
6x^2 + 27 = 0
2. Solving for x using the Quadratic Formula
Now that we have the equation in standard quadratic form, we can solve for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Where:
Substituting these values into the quadratic formula, we get:
x = (0 ± √(0^2 - 4 * 6 * 27)) / (2 * 6)
x = ± √(-648) / 12
3. Simplifying the Solution
We can simplify the solution by factoring out the square root of -1, which is represented by the imaginary unit 'i':
x = ± √(648 * -1) / 12
x = ± √648 * √-1 / 12
x = ± 18√2 * i / 12
x = ± 3√2 * i / 2
4. The Solutions
Therefore, the solutions to the quadratic equation 6x^2 + 7 = -20 are:
x = 3√2 * i / 2
x = -3√2 * i / 2
These solutions are complex numbers, meaning they involve the imaginary unit 'i'. This is because the discriminant (b^2 - 4ac) is negative, indicating that the quadratic equation has no real roots.
5. Verification
To verify our solutions, we can substitute them back into the original equation:
For x = 3√2 * i / 2:
6(3√2 * i / 2)^2 + 7 = -20
6(9 * 2 * -1 / 4) + 7 = -20
-27 + 7 = -20
-20 = -20
For x = -3√2 * i / 2:
6(-3√2 * i / 2)^2 + 7 = -20
6(9 * 2 * -1 / 4) + 7 = -20
-27 + 7 = -20
-20 = -20
Both solutions satisfy the original equation, confirming their validity.
Conclusion
In this article, we have demonstrated the step-by-step process of solving the quadratic equation 6x^2 + 7 = -20. We used the quadratic formula to find the solutions, which turned out to be complex numbers due to the negative discriminant. By verifying our solutions, we confirmed their accuracy.