Solving the Equation: 5x² + 4x – 9 = 0
To solve the quadratic equation 5x² + 4x – 9 = 0, we can use the quadratic formula:
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}},
where a, b, and c are coefficients from the equation in the form ax² + bx + c = 0.
In our case, a = 5, b = 4, and c = -9.
Step 1: Calculate the Discriminant
The first step is to calculate the discriminant (b² – 4ac):
D = b^2 - 4ac = 4^2 - 4 * 5 * (-9)
D = 16 + 180 = 196.
Since the discriminant is positive, we will have two real solutions.
Step 2: Apply the Quadratic Formula
Now, we can substitute the values into the quadratic formula:
x = \frac{{-4 \pm \sqrt{{196}}}}{{2 * 5}}.
Simplifying further:
x = \frac{{-4 \pm 14}}{{10}}.
Step 3: Calculate the Two Possible Values for x
1. For the first solution:
x_1 = \frac{{-4 + 14}}{{10}} = \frac{{10}}{{10}} = 1.
2. For the second solution:
x_2 = \frac{{-4 - 14}}{{10}} = \frac{{-18}}{{10}} = -1.8.
Final Solutions
The solutions to the equation 5x² + 4x – 9 = 0 are:
x = 1 and x = -1.8.
Conclusion
In summary, using the quadratic formula allows us to efficiently solve quadratic equations like 5x² + 4x – 9 = 0. Remember to first calculate the discriminant to determine the nature of the solutions. If the discriminant is positive, as in this case, expect two distinct real solutions.