Completing the Square: Videos & Practice Problems

Completing the square transforms any quadratic polynomial into a perfect square trinomial, enabling solutions via the square root property. By representing $x^2$ as a square and splitting the linear term's coefficient, students learn to form a binomial squared expression like ${x+3}^2$. Adjusting constants balances the equation, allowing the use of square roots to find solutions. This method deepens understanding of polynomial structure, coefficients, and exponents, enhancing problem-solving skills for any quadratic equation in standard form.

Complete the Square

Completing the square is a versatile method for solving any quadratic equation by transforming it into a perfect square trinomial. This technique involves rewriting a quadratic expression in the form x² + bx + c as a squared binomial plus or minus a constant, allowing the use of the square root property to find solutions.

To understand completing the square, visualize x² as the area of a square with side length x. The linear term, such as 6x, can be split into two equal parts (e.g., 3x + 3x) and represented as rectangles adjoining the square. By adding a smaller square in the corner with side length equal to half the coefficient of x (in this case, 3), you complete a larger square with side length x + 3. The area of this larger square is expressed algebraically as \((x + 3)^2 = x^2 + 6x + 9\).

Since the original quadratic might have a different constant term, you adjust by adding or subtracting the difference to maintain equality. For example, to rewrite \(x^2 + 6x + 12\) as a perfect square, you write it as \((x + 3)^2 + 3\), because \$9 + 3 = 12\(. This process ensures the quadratic is expressed as a perfect square plus or minus a constant.

Once the quadratic is in the form \)(x + p)^2 = q\(, solving becomes straightforward by applying the square root property: take the square root of both sides to get \)x + p = \pm \sqrt{q}\(. Then isolate x to find the solutions.

For example, solving \)x^2 + 2x - 8 = 0\( by completing the square involves first rewriting it as \)x^2 + 2x = 8\(. Half of 2 is 1, so add 1 squared (which is 1) to both sides: \)x^2 + 2x + 1 = 8 + 1\(, or \)(x + 1)^2 = 9\(. Taking the square root of both sides yields \)x + 1 = \pm 3\(, leading to solutions \)x = -1 \pm 3\(, which are \)x = 2\( and \)x = -4$.

Completing the square not only provides a method to solve any quadratic equation but also deepens understanding of the structure of quadratics by connecting algebraic expressions to geometric representations. This method is especially useful when quadratic equations are not easily factorable and prepares the foundation for deriving the quadratic formula.

Determine the value we need to add to the equation to make it a perfect square trinomial, then factor it.
$x^2+8x+$__

• A. $16;{(x+4)}^2$
• B. $8;{(x+4)}^2$
• C. $16;{(x+2)}^2$
• D. $8;{(x+2)}^2$

Answer: (A)

Determine the value we need to add to the equation to make it a perfect square trinomial, then factor it.
$y^2-10y+$__

• A. $5;{(y-5)}^2$
• B. $25;{(y+5)}^2$
• C. $25;{(y-5)}^2$
• D. $5;{(y+5)}^2$

Answer: (C)

Complete the Square Example 1

When solving quadratic equations by completing the square, special attention is needed when the leading coefficient (a) is not one or when the linear coefficient (b) is an odd number. The method can still be applied effectively by first manipulating the equation into a more manageable form.

For a quadratic equation where the coefficient of x² is not one, such as \$2x^2 - 4x - 1 = 0\(, the first step is to divide the entire equation by the leading coefficient to normalize the quadratic term. Dividing by 2 transforms the equation into \)x^2 - 2x - \frac{1}{2} = 0\(. This allows the use of the completing the square method more straightforwardly.

Completing the square involves rewriting the quadratic expression as a perfect square trinomial. Starting with \)x^2 - 2x\(, take half of the coefficient of \)x\( (which is -2), resulting in -1, and square it to get 1. Adding and subtracting this value inside the equation maintains equality. Thus, \)x^2 - 2x - \frac{1}{2}\( can be rewritten as \)(x - 1)^2 - \frac{3}{2}\(. Setting this equal to zero and solving yields:

\[(x - 1)^2 - \frac{3}{2} = 0 \\(x - 1)^2 = \frac{3}{2} \\x - 1 = \pm \sqrt{\frac{3}{2}} \\x = 1 \pm \sqrt{\frac{3}{2}}\]

This provides the two solutions for \)x\(.

When the linear coefficient \)b\( is an odd number, such as in the equation \)z^2 - 3z = 4\(, first rewrite it in standard form by subtracting 4 from both sides: \)z^2 - 3z - 4 = 0\(. Applying the completing the square method requires halving the \)b\( value, which results in fractions. Half of -3 is \)-\frac{3}{2}\(, and squaring this gives \)\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\(. Adding and subtracting \)\frac{9}{4}\( inside the equation transforms it into a perfect square trinomial:

\[z^2 - 3z - 4 = \left(z - \frac{3}{2}\right)^2 - \frac{9}{4} - 4\]

To combine the constants, rewrite -4 as \)-\frac{16}{4}\( and sum with \)-\frac{9}{4}\( to get \)-\frac{25}{4}\(. The equation becomes:

\[\left(z - \frac{3}{2}\right)^2 - \frac{25}{4} = 0\]

Solving for \)z$ involves isolating the perfect square and applying the square root property:

\[\left(z - \frac{3}{2}\right)^2 = \frac{25}{4} \\z - \frac{3}{2} = \pm \frac{5}{2} \\z = \frac{3}{2} \pm \frac{5}{2}\]

This yields two solutions:

\[z = \frac{3}{2} + \frac{5}{2} = \frac{8}{2} = 4 \\z = \frac{3}{2} - \frac{5}{2} = -\frac{2}{2} = -1\]

In summary, completing the square is a versatile method for solving quadratic equations, even when coefficients are not ideal. By normalizing the quadratic term and carefully handling fractional values when halving the linear coefficient, any quadratic can be transformed into a perfect square trinomial. This allows the application of the square root property to find exact solutions efficiently.

Solve for $x$
$8x^2-20x+12=4$

• A. $x=3,x=-\frac{1}{2}$
• B. $x=3,x=\frac{1}{2}$
• C. $x=2,x=-\frac{1}{2}$
• D. $x=2,x=\frac{1}{2}$

Answer: (D)

Complete the Square Example 2

To solve a quadratic equation by completing the square, start by visualizing the expression as a geometric square diagram to organize the terms effectively. The x² term represents the area of a large square with side lengths x. The linear term, such as −2x, is split into two equal parts, each represented as rectangles adjoining the square. For example, splitting −2x into two parts of −x each corresponds to rectangles with side lengths x and −1. Completing the square involves adding the area of a smaller square formed by the sides of length −1, which is (−1) × (−1) = 1. This transforms the quadratic expression into a perfect square trinomial: (x − 1)² = x² − 2x + 1.

However, if the original quadratic is x² − 2x + 3, the constant term differs from the perfect square trinomial by 2. To balance the equation, add 2 to both sides, rewriting the quadratic as (x − 1)² + 2. Setting this equal to zero allows solving for x:

\[(x - 1)^2 + 2 = 0\]

Subtract 2 from both sides:

\[(x - 1)^2 = -2\]

Applying the square root property, take the square root of both sides:

\[x - 1 = \pm \sqrt{-2}\]

Recognize that the square root of a negative number involves the imaginary unit i, where i = \sqrt{-1}. Using the product property of square roots:

\[\sqrt{-2} = \sqrt{2} \times \sqrt{-1} = \sqrt{2} \, i\]

Therefore, the solutions are:

\[x = 1 \pm \sqrt{2} \, i\]

This method highlights how completing the square transforms a quadratic into a perfect square form, facilitating the solution of equations with real or complex roots. Understanding the role of the imaginary unit i is essential when dealing with negative values under the square root, leading to complex solutions.

Here’s what students ask on this topic:

What is the method of completing the square in algebra?

Completing the square is a method used to solve quadratic equations by transforming any quadratic polynomial into a perfect square trinomial. This process involves rewriting the quadratic expression in the form $x^2+\; bx\; +\; c$ as ${x+d}^2+\; e$, where $d$ and $e$ are constants. The key step is to take the coefficient of the linear term, divide it by 2, and square it to find the number to add and subtract, thus creating a perfect square trinomial. This allows the quadratic to be expressed as a binomial squared, making it easier to solve using the square root property. Completing the square works for any quadratic equation and helps deepen understanding of polynomial structure.

How do you complete the square for the quadratic expression $x^2+\; 6x\; +\; 12$?

To complete the square for $x^2+\; 6x\; +\; 12$, first focus on the $x$ terms. Take the coefficient of $x$, which is 6, divide it by 2 to get 3, and then square it to get 9. Rewrite the expression as $x^2+\; 6x\; +\; 9\; +\; (12\; -\; 9)$. This becomes $(x\; +\; 3)^2\; +\; 3$. Here, $(x\; +\; 3)^2$ is the perfect square trinomial, and the +3 adjusts the constant term to keep the expression equivalent. This form allows you to solve equations by isolating the square and applying the square root property.

Why is completing the square a useful method for solving quadratic equations?

Completing the square is useful because it transforms any quadratic equation into a form that can be solved using the square root property. Unlike factoring, which only works when the quadratic factors nicely, completing the square works for all quadratics. It also provides a clear visual and algebraic understanding of the quadratic's structure by expressing it as a perfect square trinomial plus or minus a constant. This method is especially helpful when the quadratic is not easily factorable and when you want to derive the quadratic formula or analyze the vertex form of a parabola.

How do you solve the quadratic equation $x^2+\; 2x\; -\; 8\; =\; 0$ by completing the square?

To solve $x^2+\; 2x\; -\; 8\; =\; 0$ by completing the square, first move the constant to the other side: $x^2+\; 2x\; =\; 8$. Next, take half of the coefficient of $x$ (which is 2), divide by 2 to get 1, and square it to get 1. Add 1 to both sides: $x^2+\; 2x\; +\; 1\; =\; 8\; +\; 1$. This gives $(x\; +\; 1)^2\; =\; 9$. Apply the square root property: $x\; +\; 1\; =\; B1\; 3$. Finally, solve for $x$: $x\; =\; -1\; B1\; 3$, so $x\; =\; 2$ or $x\; =\; -4$.

What is the visual interpretation of completing the square?

Visually, completing the square involves representing the quadratic expression as areas of geometric shapes. The term $x^2$ is seen as the area of a square with side length $x$. The linear term, such as $6x$, is split into two rectangles each with area $3x$. These rectangles are arranged alongside the square to form an almost complete larger square. The missing small square in the corner has an area equal to the square of half the coefficient of $x$ (in this case, 9). Adding this small square completes the larger square, which corresponds algebraically to forming a perfect square trinomial. This geometric approach helps students understand why the method works.