To solve the equation x² + 4 = 7x, we first rearrange it into a standard quadratic form. This means moving all terms to one side of the equation:
x² – 7x + 4 = 0
Now, we can apply the quadratic formula, which states that for an equation of the form ax² + bx + c = 0, the solutions for x can be found using:
x = (-b ± √(b² – 4ac)) / (2a)
In our equation, a = 1, b = -7, and c = 4. Plugging these values into the formula gives:
x = (7 ± √((-7)² – 4 * 1 * 4)) / (2 * 1)
First, calculate the discriminant:
(-7)² – 4 * 1 * 4 = 49 – 16 = 33
Now, substituting this back into the formula:
x = (7 ± √33) / 2
Therefore, the exact solutions for x are:
x = (7 + √33) / 2 and x = (7 – √33) / 2
These two values are the solutions to the original equation x² + 4 = 7x.