To solve the quadratic equation 5x² + 3x + 4 = 0 using the quadratic formula, we start by recalling the formula itself. The quadratic formula is:
x = (-b ± √(b² – 4ac)) / (2a)
In our equation, we can identify the coefficients as follows:
- a = 5
- b = 3
- c = 4
Now, we can substitute these values into the quadratic formula. First, we calculate the discriminant (the part under the square root):
b² – 4ac = 3² – 4(5)(4) = 9 – 80 = -71
Since the discriminant is negative, we know that the solutions for x will be complex numbers. Now, substituting the values into the quadratic formula gives us:
x = (−3 ± √(−71)) / (2 * 5)
This simplifies to:
x = (−3 ± i√71) / 10
Here, i represents the imaginary unit. Therefore, the equation showing the quadratic formula used correctly to solve 5x² + 3x + 4 = 0 for x is:
x = (−3 ± i√71) / 10