To solve the equation x² + 6x – 25 = 9, we first need to rewrite it in a standard form by moving all terms to one side of the equation.
1. Start by subtracting 9 from both sides:
x² + 6x – 25 – 9 = 0
This simplifies to:
x² + 6x – 34 = 0
2. Now we can use the quadratic formula to find the values of x. The quadratic formula is given as:
x = (-b ± √(b² – 4ac)) / (2a)
In our equation, a = 1, b = 6, and c = -34.
3. First, we need to calculate the discriminant (b² – 4ac):
b² = 6² = 36
4ac = 4 * 1 * (-34) = -136
Now calculate the discriminant:
Discriminant = 36 – (-136) = 36 + 136 = 172
4. Next, plug the values of a, b, and the discriminant into the quadratic formula:
x = (-6 ± √172) / (2 * 1)
5. Simplifying further:
x = (-6 ± √172) / 2
6. We can simplify √172:
√172 = √(4 * 43) = 2√43
7. Substituting back in gives:
x = (-6 ± 2√43) / 2
8. Dividing everything by 2:
x = -3 ± √43
Thus, the solutions to the equation are:
x = -3 + √43 and x = -3 – √43
To get numerical approximations, you can calculate:
- x ≈ -3 + 6.56 ≈ 3.56
- x ≈ -3 – 6.56 ≈ -9.56
In conclusion, the values of x in the given equation are approximately:
- x ≈ 3.56
- x ≈ -9.56