key term - Completing the Square

Definition

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square form. This technique allows for the application of the square root property or the quadratic formula to find the solutions to the equation.

5 Must Know Facts For Your Next Test

1. Completing the square is a technique used to transform a quadratic equation in standard form ($ax^2 + bx + c = 0$) into a perfect square form ($a(x - h)^2 + k = 0$).
2. The process involves adding and subtracting a constant to both sides of the equation to create a perfect square on the left-hand side.
3. Completing the square is a crucial step in solving quadratic equations using the square root property or the quadratic formula.
4. The constant added to both sides of the equation is $\frac{b^2}{4a}$, which ensures that the left-hand side becomes a perfect square.
5. Completing the square can be used to determine the vertex of a parabolic function, as the vertex is located at the point $(-\frac{b}{2a}, \frac{b^2 - 4ac}{4a})$.

Review Questions

• Explain the purpose of completing the square and how it relates to solving quadratic equations.
  • The purpose of completing the square is to transform a quadratic equation in standard form ($ax^2 + bx + c = 0$) into a perfect square form ($a(x - h)^2 + k = 0$). This transformation allows for the application of the square root property or the quadratic formula to find the solutions to the equation. By completing the square, the equation is rearranged into a form that makes it easier to solve, as the solutions can be directly obtained from the perfect square form or by using the quadratic formula.
• Describe the step-by-step process of completing the square for a quadratic equation in standard form.
  • To complete the square for a quadratic equation in standard form ($ax^2 + bx + c = 0$), follow these steps: 1. Divide both sides of the equation by the coefficient of $x^2$, $a$, to get the equation in the form $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$. 2. Add the square of half the coefficient of $x$, $\frac{b}{2a}$, to both sides of the equation to create a perfect square on the left-hand side: $x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a}$. 3. Factor the left-hand side to obtain the perfect square form: $(x + \frac{b}{2a})^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a}$.
• Explain how completing the square can be used to determine the vertex of a parabolic function and discuss the significance of this information.
  • When a quadratic equation is in the form $ax^2 + bx + c = 0$, completing the square can be used to determine the vertex of the corresponding parabolic function. The vertex is located at the point $(-\frac{b}{2a}, \frac{b^2 - 4ac}{4a})$. This information is significant because the vertex represents the minimum or maximum point of the parabolic function, which is crucial for understanding the behavior and characteristics of the function. Knowing the vertex allows for the accurate graphing and analysis of the parabolic function, as well as the identification of its key features, such as the axis of symmetry, range, and domain.

Related terms

Quadratic Equation:

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

Square Root Property:

The property that states if $x^2 = a$, then $x = \pm \sqrt{a}$.

Quadratic Formula:

The formula used to solve any quadratic equation in the form $ax^2 + bx + c = 0$, given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.