A quadratic equation is a polynomial equation of the second degree. Its standard form is a x squared plus b x plus c equals zero, where a, b, and c are coefficients, and a cannot be zero. The graph of a quadratic equation forms a parabola, which can open upward or downward depending on the sign of coefficient a.
Quadratic equations can be classified into three types based on their discriminant, which is b squared minus four a c. When the discriminant is positive, the equation has two distinct real roots and the parabola intersects the x-axis at two points. When the discriminant equals zero, there is exactly one real root and the parabola touches the x-axis at one point. When the discriminant is negative, there are no real roots and the parabola does not intersect the x-axis.
The quadratic formula is a universal method to solve any quadratic equation. The formula is x equals negative b plus or minus the square root of b squared minus four a c, all divided by two a. To use this formula, first identify the coefficients a, b, and c from the standard form. Then calculate the discriminant. Finally, substitute the values into the formula. For example, solving x squared minus five x plus six equals zero, we get a equals one, b equals negative five, and c equals six. The discriminant is one, so the solutions are x equals three or x equals two.
Besides the quadratic formula, there are other methods to solve quadratic equations. The factoring method involves finding two numbers that multiply to give a c and add to give b, then expressing the equation as a product of two binomials. Completing the square transforms the equation into the form x plus h squared equals k. The graphing method finds solutions by identifying where the parabola crosses the x-axis. Each method has its advantages depending on the specific equation you need to solve.
To summarize what we have learned about quadratic equations: They are second-degree polynomial equations with the standard form a x squared plus b x plus c equals zero. The discriminant determines how many real solutions exist. We can solve them using various methods including the quadratic formula, factoring, and completing the square. Quadratic equations have wide applications in physics, engineering, and many areas of mathematics.