Overview
A quadratic equation can have two, one or no real roots. First move all terms to one side so the zero-product property applies to a product equal to zero. Factoring is efficient when coefficients allow it; completing a square or using the quadratic formula handles other cases.
Consider this example: What is the larger solution of x² − 7x + 10 = 0? Factor to (x − 2)(x − 5) = 0. The solutions are 2 and 5, so the larger is 5. The number 7 is the sum of the roots, not a root.
The method also handles a different case: Solve (x − 3)² = 16. What is the smaller solution? Taking square roots gives x − 3 = 4 or −4. Thus x = 7 or −1, and the smaller root is −1. Using only the positive square root loses that solution.
For a further application, consider: For which real value of k does x² − 6x + k = 0 have exactly one real solution? Complete the square: (x − 3)² = 9 − k. Exactly one real root occurs when the right side is zero, so k = 9 and x = 3. If k < 9 there are two roots; if k > 9 there are no real roots.
Setting a factor to zero is valid when the entire product equals zero, not when it equals another number. Squaring produces two signs when recovering a real square root, unless the root is zero. Check every candidate in the original equation. On a new quadratic, choose the shortest method and keep domain restrictions from any original denominator or stated context.
Completing the square is useful when an integer factorization is unavailable. Solve x² + 4x + 1 = 0 by moving the constant: x² + 4x = −1. Half the linear coefficient is 2; its square is 4, which must be added to both sides. Then (x + 2)² = 3. Taking both square roots gives x + 2 = ±√3 and x = −2 ± √3. The center is −2, not +2, because 2 is subtracted after taking roots. To check exactly, use x² + 4x + 1 = (x + 2)² − 3: either proposed root makes this 3 − 3 = 0. Keeping only one sign loses a real solution. This also connects the square method to counting roots: the sign of the square’s right side tells whether two, one, or no real solutions exist.