Solve for x using the quadratic formula.
(Type an exact answer, using radicals as needed. Express complex numbers in terms of i. Use a comma to separate answers as needed.
4x^2 = 10x - 9
To solve the quadratic equation 4x^2 = 10x - 9, we can rearrange it to the standard form and then use the quadratic formula, which is given by:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, a = 4, b = -10, and c = 9. Substituting these values into the quadratic formula, we have:
x = (-(-10) ± √((-10)^2 - 4(4)(9))) / (2(4))
= (10 ± √(100 - 144)) / 8
= (10 ± √(-44)) / 8
Since the value inside the square root is negative, we can rewrite it as √(-1 * 4 * 11) = √(4 * 11) * i = 2√11 i.
Therefore, the solutions for x are:
x = (10 + 2√11 i) / 8
x = (10 - 2√11 i) / 8
Simplifying further:
x = (5 + √11 i) / 4
x = (5 - √11 i) / 4
Hence, the exact solutions for x are:
x = (5 + √11 i) / 4, (5 - √11 i) / 4