To find the solutions of the equation 4x² + 3x – 24 = 0, we can use the quadratic formula. The standard form of a quadratic equation is ax² + bx + c = 0, where a = 4, b = 3, and c = -24.
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / (2a)
First, we need to calculate the discriminant (b² – 4ac):
- b² = 3² = 9
- 4ac = 4 * 4 * (-24) = -384
- Discriminant = 9 – (-384) = 9 + 384 = 393
Now, we can substitute the values into the quadratic formula:
x = ( -3 ± √393 ) / ( 2 * 4 )
x = ( -3 ± √393 ) / 8
The solutions to the equation are thus:
- x₁ = ( -3 + √393 ) / 8
- x₂ = ( -3 – √393 ) / 8
These two values of x represent the points where the parabola intersects the x-axis. Thus, we have found the solutions to our equation.