x2 + bx + c and ax2 + bx + c.
The expressions you mentioned are quadratic equations. A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. These equations are commonly written in the form ax^2 + bx + c, where a, b, and c are constants.
To solve a quadratic equation, we need to find the values of x that make the equation equal to zero. There are different methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.
1. Factoring: If the quadratic equation can be factored, we can set each factor equal to zero and solve for x. For example, for the equation x^2 + bx + c = 0, we can factor it into (x + p)(x + q) = 0, where p and q are the factors. Then we can set each factor equal to zero: x + p = 0 and x + q = 0, and solve for x.
2. Completing the square: If factoring is not possible or convenient, we can use the method of completing the square. We rewrite the equation in the form (x + h)^2 = k, where h and k are constants. By rearranging the terms and solving for x, we can find the solutions.
3. Quadratic formula: The quadratic formula is a formula that gives the solutions of any quadratic equation. It is derived by completing the square. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a). By substituting the values of a, b, and c from the equation ax^2 + bx + c = 0 into the quadratic formula, we can find the solutions for x.
Each method has its merits and is useful in different scenarios. The choice of method depends on the specific equation and the desired approach.