A quadratic equation is an algebraic equation where the highest power of the variable is 2. The standard form is ax² + bx + c = 0, where a, b, and c are coefficients and a cannot be zero. The value of a controls the opening direction and width of the parabola, b affects its horizontal position, and c shows where the graph crosses the y-axis.

Quadratic equations appear in algebra, geometry, physics, economics, engineering, and optimization problems. The roots are the x-values where the expression becomes zero. On a graph, those roots are the points where the parabola crosses or touches the x-axis.

The quadratic formula solves any equation written in the form ax² + bx + c = 0. The part inside the square root, b² − 4ac, is called the discriminant. It tells you what type of answer to expect before calculating the roots.

Discriminant

The value b² − 4ac. If it is positive, the equation has two real roots. If it is zero, the equation has one repeated real root. If it is negative, the equation has two complex conjugate roots.

The graph of a quadratic equation is a parabola. When a is positive, the parabola opens upward and has a minimum point. When a is negative, it opens downward and has a maximum point. This turning point is called the vertex.

The x-coordinate of the vertex is calculated using −b ÷ 2a. The y-coordinate is found by substituting that x-value back into ax² + bx + c. The vertex helps explain the symmetry of the graph and shows the highest or lowest value of the expression.

• Check the sign of each coefficient before solving.
• Remember that x² − 5x + 6 has b = −5, not 5.
• Use both the plus and minus branches of the formula.
• Do not treat complex roots as graph x-intercepts.
• Confirm that a is not zero before using the quadratic formula.