To solve the equation x² + 4x + 9 = 29, follow these steps:
Step 1: Rearrange the equation. Start by moving all terms to one side to set the equation to zero. We can do this by subtracting 29 from both sides:
x² + 4x + 9 - 29 = 0
Which simplifies to:
x² + 4x - 20 = 0
Step 2: Identify coefficients. In the quadratic equation ax² + bx + c = 0, we have:
- a = 1 (coefficient of x²)
- b = 4 (coefficient of x)
- c = -20 (the constant term)
Step 3: Use the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
In our case, substituting the values of a, b, and c:
x = (–4 ± √(4² - 4 * 1 * (–20))) / (2 * 1)
Step 4: Calculate the discriminant. First calculate the part under the square root:
b² - 4ac = 16 + 80 = 96
Step 5: Solve for x. Plugging this back into the quadratic formula:
x = (–4 ± √96) / 2
Since √96 can be simplified:
√96 = √(16 * 6) = 4√6
Now substituting this back gives:
x = (–4 ± 4√6) / 2
This simplifies to:
x = –2 ± 2√6
Step 6: State the final solutions. Therefore, the solutions for x are:
- x = –2 + 2√6
- x = –2 – 2√6
In conclusion, we have found the two possible values for x that satisfy the original equation.