key term - Complete the Square

Definition

Completing the square is a mathematical technique used to transform a quadratic expression in the form $ax^2 + bx + c$ into the standard form $a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. This method is particularly useful when solving quadratic equations and graphing parabolas.

5 Must Know Facts For Your Next Test

1. Completing the square is a useful technique for solving quadratic equations, as it allows you to find the roots or zeros of the equation.
2. The process of completing the square involves adding and subtracting a constant to the original expression to make it a perfect square.
3. Completing the square is also essential for graphing parabolas, as it allows you to easily identify the vertex and axis of symmetry of the parabola.
4. The standard form of a quadratic expression, $a(x - h)^2 + k$, is obtained by completing the square, where $(h, k)$ represents the vertex of the parabola.
5. Completing the square is a key concept in the rotation of axes, as it is used to transform the equation of a conic section (such as a parabola) into its standard form.

Review Questions

• Explain the process of completing the square and how it can be used to solve quadratic equations.
  • To complete the square, you start with a quadratic expression in the form $ax^2 + bx + c$. First, you divide the coefficient of $x$ (the $b$ term) by twice the coefficient of $x^2$ (the $a$ term), which gives you $h = b/(2a)$. Then, you add the square of this value, $h^2 = (b/(2a))^2$, to both sides of the equation. This transforms the original expression into the standard form $a(x - h)^2 + k$, where $k = c - b^2/(4a)$. Solving the resulting equation in standard form allows you to find the roots or zeros of the quadratic equation.
• Describe how completing the square is used in the context of rotating the axes of a parabola.
  • When rotating the axes of a parabola, completing the square is a crucial step. By transforming the equation of the parabola into the standard form $a(x - h)^2 + k$, you can easily identify the vertex $(h, k)$ and the axis of symmetry of the parabola. This information is then used to determine the angle of rotation and the new coordinates of the parabola after the rotation. Completing the square allows you to simplify the equation and make the process of rotating the axes more straightforward.
• Analyze how the process of completing the square is related to the properties of parabolas, such as the vertex and axis of symmetry.
  • By completing the square, you are essentially rearranging the quadratic expression into the form $a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. This transformation is significant because the vertex is the point where the parabola changes from decreasing to increasing (or vice versa), and it corresponds to the maximum or minimum value of the function. Additionally, the axis of symmetry of the parabola passes through the vertex and is perpendicular to the $x$-axis. Therefore, the process of completing the square is directly linked to the key properties of parabolas, allowing you to easily identify and work with these important characteristics.

Related terms

Quadratic Expression: A polynomial expression of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

Standard Form:

The form of a quadratic expression written as $a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola.

Vertex:

The point on a parabola where the graph changes from decreasing to increasing (or vice versa), which corresponds to the maximum or minimum value of the function.