分步说明
Write Down the Quadratic Equation
Start by writing down the given quadratic equation in standard form: $ax^2 + bx + c = 0$. Identify the values of $a$, $b$, and $c$.
Move the Constant Term to the Right-Hand Side
Move the constant term to the right-hand side of the equation: $ax^2 + bx = -c$. This step is necessary to isolate the terms with the variable $x$.
Factor Out the Coefficient of $x^2$
Factor out the coefficient of $x^2$ from the left-hand side of the equation: $a(x^2 + rac{b}{a}x) = -c$. This step is necessary to create a perfect square trinomial.
Add and Subtract the Square of Half the Coefficient of $x$
Add and subtract the square of half the coefficient of $x$ inside the parentheses: $a(x^2 + rac{b}{a}x + ( rac{b}{2a})^2 - ( rac{b}{2a})^2) = -c$. This step is necessary to create a perfect square trinomial.
Simplify the Equation
Simplify the equation by combining like terms: $a(x + rac{b}{2a})^2 - rac{b^2}{4a} = -c$. Then, add $ rac{b^2}{4a}$ to both sides of the equation to get $a(x + rac{b}{2a})^2 = -c + rac{b^2}{4a}$. This is the vertex form of the quadratic equation.
Write the Final Answer in Vertex Form
Write the final answer in vertex form: $a(x - h)^2 + k = 0$, where $h = - rac{b}{2a}$ and $k = c - rac{b^2}{4a}$. This is the solution to the quadratic equation.
Introduction to Completing the Square
Completing the square is a method used to solve quadratic equations. It involves transforming the standard form of a quadratic equation into vertex form. The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The vertex form of a quadratic equation is $a(x - h)^2 + k = 0$, where $(h, k)$ is the vertex of the parabola.
The Completing the Square Formula
The formula for completing the square is $a(x - h)^2 + k = 0$, where $h = - rac{b}{2a}$ and $k = c - rac{b^2}{4a}$. To complete the square, we need to move the constant term to the right-hand side of the equation and then group the like terms.
Step-by-Step Guide to Completing the Square
The following steps will guide you through the process of completing the square: