Which of the following equations has only one solution?
x² = 9, x(x - 1) = 9, x² - 6x + 9 = 0

Solution:

To find which equation has only one solution, we will solve for x in each of the given equations.

Consider x² = 9

⇒ x² - 9 = 0 or x² - (3)²

⇒ (x + 3)(x - 3) [using the algebraic identity a² - b² = (a + b)(a - b)]

⇒ x = -3 or x = 3

Thus, this equation has two solutions.

x(x - 1) = 9

Simplify the equation.

⇒ x² - x - 9 = 0

Using quadratic formula, where a = 1, b = -1 and c = -9

x = - 1 ± √ [(-1)² - 4 (1)(-9)] / 2(1)

x = -1 ± √ 1 + 36 / 2

x = - 1 - √ 37/ 2 or -1 + √ 37/ 2 

Thus, this equation has two solutions.

Consider x² - 6x + 9 = 0

Factorize the quadratic equation by splitting the middle term.

x² - 3x - 3x + 9 = 0

x(x - 3) -3(x - 3) = 0

(x - 3)(x - 3) = 0

x = 3

Thus, this equation has only one solution.

x² - 6x + 9 = 0 has only one solution.

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Which of the following equations has only one solution?

Summary:

The equation x² - 6x + 9 = 0 has only one solution x = 3.