Completing The Square

Completing the square is a mathematical technique used to transform a quadratic equation into a perfect square. This process allows for easier solving of the equation and finding its solutions.

Steps to Complete the Square

To complete the square for a quadratic equation of the form $$ax^2 + bx + c = 0$$, follow these steps:

1. Divide both sides of the equation by the coefficient of $x^2$. This will give you an equation in the form $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$.

2. Move the constant term $\frac{c}{a}$ to the other side of the equation. This will give you an equation in the form $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$.

3. Take half of the coefficient of $x$ and square it. This will give you the value of $\left(\frac{b}{2a}\right)^2$.

4. Add $\left(\frac{b}{2a}\right)^2$ to both sides of the equation. This will give you an equation in the form $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$.

5. Factor the left side of the equation. This will give you an equation in the form $$(x + \frac{b}{2a})^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$.

6. Take the square root of both sides of the equation. This will give you an equation in the form $$x + \frac{b}{2a} = \pm\sqrt{-\frac{c}{a} + \left(\frac{b}{2a}\right)^2}$$.

7. Solve for $x$. This will give you the solutions to the quadratic equation.

Example

To illustrate the process of completing the square, let’s consider the quadratic equation $$2x^2 + 4x - 5 = 0$$.

1. Divide both sides of the equation by 2. This gives us $$x^2 + 2x - \frac{5}{2} = 0$$.

2. Move the constant term $-5/2$ to the other side of the equation. This gives us $$x^2 + 2x = \frac{5}{2}$$.

3. Take half of the coefficient of $x$ and square it. This gives us $(2/2)^2 = 1$.

4. Add 1 to both sides of the equation. This gives us $$x^2 + 2x + 1 = \frac{5}{2} + 1$$.

5. Factor the left side of the equation. This gives us $$(x + 1)^2 = \frac{7}{2}$$.

6. Take the square root of both sides of the equation. This gives us $$x + 1 = \pm\sqrt{\frac{7}{2}}$$.

7. Solve for $x$. This gives us $$x = -1 \pm\sqrt{\frac{7}{2}}$$.

Therefore, the solutions to the quadratic equation $$2x^2 + 4x - 5 = 0$$ are $$x = -1 + \sqrt{\frac{7}{2}}$$ and $$x = -1 - \sqrt{\frac{7}{2}}$$.

Completing the Square Formula

The completing the square formula is a mathematical technique used to solve quadratic equations. It involves manipulating the equation into a form that makes it easier to find the solutions, or roots, of the equation.

Steps to Complete the Square

To complete the square for a quadratic equation of the form $$ax^2 + bx + c = 0$$, follow these steps:

1. Divide both sides of the equation by the coefficient of $x^2$. This will give you an equation in the form $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$.

2. Move the constant term $\frac{c}{a}$ to the other side of the equation. This will give you an equation in the form $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$.

3. Take half of the coefficient of $x$, square it, and add it to both sides of the equation. This will give you an equation in the form $$(x + \frac{b}{2a})^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$.

4. Simplify the right side of the equation. This will give you an equation in the form $$(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}$$.

5. Take the square root of both sides of the equation. This will give you two equations in the form $$x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}$$.

6. Solve each equation for $x$. This will give you the two solutions, or roots, of the quadratic equation.

Example

To illustrate the steps of completing the square, let’s solve the quadratic equation $$2x^2 + 4x - 5 = 0$$.

1. Divide both sides of the equation by 2. This gives us $$x^2 + 2x - \frac{5}{2} = 0$$.

2. Move the constant term $-5/2$ to the other side of the equation. This gives us $$x^2 + 2x = \frac{5}{2}$$.

3. Take half of the coefficient of $x$, square it, and add it to both sides of the equation. This gives us $$(x + 1)^2 = \frac{5}{2} + 1 = \frac{7}{2}$$.

4. Simplify the right side of the equation. This gives us $$(x + 1)^2 = \frac{7}{2}$$.

5. Take the square root of both sides of the equation. This gives us $$x + 1 = \pm \sqrt{\frac{7}{2}}$$.

6. Solve each equation for $x$. This gives us $$x = -1 \pm \sqrt{\frac{7}{2}}$$.

Therefore, the solutions to the quadratic equation $$2x^2 + 4x - 5 = 0$$ are $$x = -1 + \sqrt{\frac{7}{2}}$$ and $$x = -1 - \sqrt{\frac{7}{2}}$$.

Steps to Solve Quadratic Equations using Completing the Square

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

Completing the square is a method for solving quadratic equations that involves transforming the equation into the form $(x - h)^2 = k$, where $h$ and $k$ are real numbers.

Here are the steps to solve a quadratic equation using completing the square:

1. Move the constant term to the other side of the equation.

$$ax^2 + bx = -c$$

2. Divide both sides of the equation by the coefficient of $x^2$.

$$x^2 + \frac{b}{a}x = -\frac{c}{a}$$

3. Add the square of half the coefficient of $x$ to both sides of the equation.

$$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$

4. Factor the left side of the equation.

$$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$

5. Take the square root of both sides of the equation.

$$x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a} + \left(\frac{b}{2a}\right)^2}$$

6. Solve for $x$.

$$x = -\frac{b}{2a} \pm \sqrt{-\frac{c}{a} + \left(\frac{b}{2a}\right)^2}$$

Example:

Solve the quadratic equation $x^2 - 4x - 5 = 0$.

Solution:

1. Move the constant term to the other side of the equation.

$$x^2 - 4x = 5$$

2. Divide both sides of the equation by the coefficient of $x^2$.

$$x^2 - 4x = 5$$

3. Add the square of half the coefficient of $x$ to both sides of the equation.

$$x^2 - 4x + 4 = 5 + 4$$

4. Factor the left side of the equation.

$$(x - 2)^2 = 9$$

5. Take the square root of both sides of the equation.

$$x - 2 = \pm \sqrt{9}$$

6. Solve for $x$.

$$x = 2 \pm 3$$

Therefore, the solutions to the equation $x^2 - 4x - 5 = 0$ are $x = 5$ and $x = -1$.

Geometric Interpretation for Completing the Square Method

The completing the square method is a technique used to solve quadratic equations. It involves adding and subtracting a constant term to the equation in order to transform it into a perfect square. This geometric interpretation provides a visual representation of the process and helps in understanding the underlying concept.

Key Concepts:

• Parabola: A parabola is a U-shaped curve that is defined by the equation $y = ax^2 + bx + c$. The vertex of a parabola is the point where it changes direction.

• Vertex Form: The vertex form of a quadratic equation is given by $y = a(x - h)^2 + k$, where $(h, k)$ are the coordinates of the vertex.

Geometric Interpretation:

1. Starting Point: Consider a quadratic equation in the form of $y = ax^2 + bx + c$. The graph of this equation is a parabola.

2. Shifting the Parabola: To complete the square, we add and subtract a constant term $\left(\frac{b}{2a}\right)^2$ to the equation. This shifts the parabola horizontally without changing its shape.

3. Creating a Perfect Square: The added term $\left(\frac{b}{2a}\right)^2$ ensures that the expression inside the parentheses becomes a perfect square. This transforms the equation into the vertex form $y = a(x - h)^2 + k$.

4. Vertex: The vertex $(h, k)$ of the parabola represents the minimum or maximum point of the graph. The value of $h$ is given by $-\frac{b}{2a}$, and the value of $k$ is determined by substituting $h$ back into the original equation.

5. Axis of Symmetry: The line $x = h$ is the axis of symmetry for the parabola. It divides the parabola into two symmetrical halves.

The geometric interpretation of the completing the square method provides a visual understanding of how the process transforms a quadratic equation into a perfect square. By shifting the parabola horizontally and creating a perfect square, we can easily identify the vertex and axis of symmetry of the parabola, which are crucial for solving quadratic equations and understanding their behavior.

Importance of Completing the Square Method in Quadratic Equations

Completing the square method is a technique used to solve quadratic equations. It involves transforming a quadratic equation into a perfect square, making it easier to find the solutions or roots of the equation. This method is particularly useful when dealing with quadratic equations that cannot be easily solved using other methods like factoring or using the quadratic formula.

Steps Involved in Completing the Square Method

1. Move the constant term to the other side of the equation:

   • Start by moving the constant term (the number without a variable) to the other side of the equation. This will ensure that the quadratic term (the term with the squared variable) is on one side of the equation.

2. Divide both sides of the equation by the coefficient of the squared variable:

   • Divide both sides of the equation by the coefficient of the squared variable. This will make the coefficient of the squared variable equal to 1.

3. Add the square of half the coefficient of the linear variable to both sides of the equation:

   • Find half the coefficient of the linear variable (the term with the variable without a square).
   • Square this value and add it to both sides of the equation.

4. Factor the left side of the equation:

   • The left side of the equation should now be a perfect square. Factor it into the form $(x + a)^2$.

5. Set the factored expression equal to zero and solve for x:

   • Set the factored expression equal to zero and solve for the variable x. This will give you the solutions or roots of the quadratic equation.

Advantages of Completing the Square Method

• Simplicity: Completing the square method is relatively simple to understand and apply, making it accessible to students of all levels.

• Wide Applicability: This method can be used to solve a wide range of quadratic equations, including those that cannot be easily solved by factoring or using the quadratic formula.

• Accuracy: Completing the square method provides exact solutions to quadratic equations, avoiding the approximations that can occur with other methods.

• Geometric Interpretation: This method offers a geometric interpretation of quadratic equations, helping students visualize the solutions as points on a graph.

Completing the square method is an important technique for solving quadratic equations. Its simplicity, wide applicability, accuracy, and geometric interpretation make it a valuable tool for students and mathematicians alike. By understanding and mastering this method, individuals can effectively solve a variety of quadratic equations and gain a deeper understanding of their behavior and properties.