Find the roots of the quadratic equation \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = 5 \), and \( c = -3 \).
To find the roots of a quadratic equation, we use the quadratic formula:
$$ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} $$
Where:
Given:
Substitute these values into the quadratic formula:
$$ x = \frac{{-5 \pm \sqrt{{5^2 - 4 \cdot 2 \cdot (-3)}}}}{{2 \cdot 2}} $$
Calculate the discriminant:
$$ 5^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49 $$
Substitute back into the formula:
$$ x = \frac{{-5 \pm \sqrt{{49}}}}{{4}} $$
$$ x = \frac{{-5 \pm 7}}{{4}} $$
Therefore, the roots are:
$$ x_1 = \frac{{-5 + 7}}{{4}} = \frac{2}{4} = 0.5 $$
$$ x_2 = \frac{{-5 - 7}}{{4}} = \frac{-12}{4} = -3 $$
So, the roots of the quadratic equation \( 2x^2 + 5x - 3 = 0 \) are \( x_1 = 0.5 \) and \( x_2 = -3 \).
Now, let's solve this problem using Python. We will define a function to calculate the roots of a quadratic equation.
```python
import math
def solve_quadratic(a, b, c):
# Calculate the discriminant
discriminant = b**2 - 4*a*c
# Check if roots are real
if discriminant < 0:
return "No real roots"
# Calculate the roots
sqrt_discriminant = math.sqrt(discriminant)
x1 = (-b + sqrt_discriminant) / (2*a)
x2 = (-b - sqrt_discriminant) / (2*a)
return x1, x2
# Given coefficients
a = 2
b = 5
c = -3
# Calculate the roots
roots = solve_quadratic(a, b, c)
print(f"The roots of the quadratic equation {a}x^2 + {b}x + {c} = 0 are {roots}")
We define a function solve_quadratic that takes coefficients \( a \), \( b \), and \( c \) as inputs. It calculates the discriminant and then determines the roots based on the discriminant's value.
We assign the given values of \( a \), \( b \), and \( c \).
We call the solve_quadratic function with the given coefficients and print the roots.
By following these steps, we've solved a quadratic equation problem both manually and programmatically. Explore more math challenges and solutions on SkoolTest!