To solve the quadratic equation x^2 + 9x - 5 = 0, you can use the quadratic formula or complete the square method. I'll show you both methods:
Method 1: Quadratic Formula
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In your equation, a = 1, b = 9, and c = -5. Substituting these values into the quadratic formula, we have:
x = (-9 ± √(9^2 - 4 * 1 * -5)) / (2 * 1)
Simplifying further:
x = (-9 ± √(81 + 20)) / 2
x = (-9 ± √101) / 2
Therefore, the solutions to the equation x^2 + 9x - 5 = 0 are:
x = (-9 + √101) / 2
x = (-9 - √101) / 2
Method 2: Completing the Square
To use the completing the square method, we rewrite the equation in the form (x + p)^2 = q. Here's how it can be done:
1. Move the constant term (-5) to the other side of the equation:
x^2 + 9x = 5
2. To complete the square, take half of the coefficient of x (9/2) and square it:
(9/2)^2 = 81/4
3. Add this value to both sides of the equation:
x^2 + 9x + 81/4 = 5 + 81/4
x^2 + 9x + 81/4 = 20/4 + 81/4
x^2 + 9x + 81/4 = 101/4
4. Rewrite the left side of the equation as a perfect square:
(x + 9/2)^2 = 101/4
5. Take the square root of both sides (don't forget the ±):
x + 9/2 = ±√(101/4)
6. Solve for x:
x = -9/2 ± √(101/4)
Simplifying further:
x = (-9 ± √101) / 2
So, the solutions to the equation x^2 + 9x - 5 = 0 are the same as we obtained using the quadratic formula:
x = (-9 + √101) / 2
x = (-9 - √101) / 2
Both methods yield the same solutions.