A worked example: solving 3x − 9 = 0
For the linear equation 3x − 9 = 0, the coefficients are a = 3 and b = −9. Applying x = −b ÷ a gives x = −(−9) ÷ 3 = 3. Substituting back, 3 × 3 − 9 = 0, confirming the root.
The formula: linear and quadratic equations
A linear equation with two coefficients, ax + b = 0, always solves as x = −b ÷ a, provided a is not zero. A quadratic equation with three coefficients, ax² + bx + c = 0, needs the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 3x + 2 = 0 (a = 1, b = −3, c = 2), the discriminant b² − 4ac = 9 − 8 = 1, a positive number, so there are two distinct real roots: x = (3 + 1) ÷ 2 = 2 and x = (3 − 1) ÷ 2 = 1.
Another worked example: a repeated root
For x² − 14x + 49 = 0 (a = 1, b = −14, c = 49), the discriminant is (−14)² − 4 × 1 × 49 = 196 − 196 = 0. A zero discriminant means the parabola just touches the x-axis at a single point rather than crossing it twice, so the equation has one repeated root: x = 14 ÷ 2 = 7. Substituting back confirms it: 7² − 14 × 7 + 49 = 49 − 98 + 49 = 0.
Why the discriminant matters before you even solve
The discriminant, b² − 4ac, tells you what kind of answer to expect before doing the rest of the arithmetic. Positive means two distinct real roots, zero means one repeated real root, and negative means the two roots are a complex-conjugate pair (real part ± imaginary part), since no real number squares to a negative value under the square root. Checking its sign first avoids being surprised by an unexpected answer format.