How to Solve a Quadratic Equation by Completing the Square?

Completing the Square is a way used to solve a quadratic equation by changing the form of the equation. In this step-by-step guide, you learn more about the method of completing the square.

How to Solve a Quadratic Equation by Completing the Square?

When we want to convert a quadratic expression of the form \(ax^2+ bx+c\) to the vertex form, we use the completing the square method.

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Step by Step guide to completing the square

Completing a square is a method used to convert a quadratic expression of the form \(ax^2+ bx+c\) to the vertex form \(a(x-h)^2+k\). The most common application of completing the square is in solving a quadratic equation. This can be done by rearranging the expression obtained after completing the square: \(a(x + m)^2+n\), so that the left side is a perfect square trinomial.

Completing the square method is useful in the following cases:

  • Converting a quadratic expression into vertex form.
  • Analyzing at which point the quadratic expression has minimum or maximum value.
  • Graphing a quadratic function.
  • Solving a quadratic equation.
  • Deriving the quadratic formula.

Completing the square method

The most common application of completing the square method is factorizing a quadratic equation, and henceforth finding the roots and zeros of a quadratic polynomial or a quadratic equation. We know that a quadratic equation in the form of \(ax^2+bx+c=0\) can be solved by the factorization method. But sometimes, factorization of the quadratic expression \(ax^2+bx+c\) is complex or impossible.

Completing the square formula

Completing a square formula is a method for converting a quadratic polynomial or equation to a complete square with an additional constant value. A quadratic expression in variable \(x\): \(ax^2+ bx+ c\), where \(a, b\) and \(c\) are any real numbers but \(a≠0\), can be converted into a perfect square with some additional constant by using completing the square formula or technique.

Completing the square formula is a method that can be used to find the roots of the given quadratic equations, \(ax^2+bx+c\), where \(a, b\) and \(c\) are any real numbers but \(a≠0\).

Formula for completing the square:

The formula for completing the square is:

\(\color{blue}{ax^2 + bx + c ⇒ a(x + m)^2+ n}\)

where:

  • \(m\) is any real number
  • \(n\) is a constant term

Instead of using the complicated step-by-step method to complete the square, we can use the simple formula below to complete the square. To complete the square in the expression \(ax^2+bx + c\), first find:

\(\color{blue}{m= \frac{b}{2a}}\) , \(\color{blue}{n=c – (\frac{b^2}{4a})}\)

Substitute these values in: \(ax^2+bx +c = a(x + m)^2+n\). These formulas are derived geometrically.

Solving quadratic equations using completing the square method

Let’s complete the square in the expression \(ax^2+bx+c\) using the square and rectangle in Geometry. The coefficient of \(x^2\) must be made \(1\) by taking \(a\) as the common factor. We get,

\(ax^2+bx+c= a(x^2+\frac{b}{a}x+\frac{c}{a})\)→\((1)\)

Now, consider the first two expressions \(x^2\) and \((\frac{b}{a}) x\). Let us consider a square of side \(x\) (whose area is \(x^2\)). Also consider a rectangle of length \((\frac{b}{a})\) and breadth \((x)\) (whose area is \((\frac{b}{a})x)\).

Now, divide the rectangle into two equal parts. The length of each rectangle will be \(\frac{b}{2a}\).

Attach half of this rectangle to the right side of the square and the remaining half to the bottom of the square.

To complete a geometric square, there is some shortage which is a square of side \(\frac{b}{2a}\). The square of the area \([(\frac{b}{2a})^2]\) should be added to \(x^2+ (\frac{b}{a})x\) to complete the square.

But, we cannot just add, we need to subtract it to retain the expression’s value. Therefore, to complete the square:

\(x^2+ (\frac{b}{a})x= x^2+ (\frac{b}{a})x + (\frac{b}{2a})^2 – (\frac{b}{2a})^2\)
\(=x^2+ (\frac{b}{a})x+(\frac{b}{2a})^2 – \frac{b^2}{4a^2}\)

Multiplying and dividing \((\frac{b}{a})x\) with \(2\) gives:

\(x^2 + (\frac{2⋅x⋅b}{2a}) + (\frac{b}{2a})^2 – \frac{b^2}{4a^2}\)

By using the identity, \(x^2+ 2xy + y^2 = (x + y)^2\)
The above equation can be written as,
\(x^2+ bax = (x + \frac{b}{2a})^2 – (\frac{b^2}{4a^2})\)

By substituting this in \((1)\):

\(ax^2+bx+c = a((x + \frac{b}{2a})^2 – \frac{b^2}{4a^2}+\frac{c}{a})= a(x + \frac{b}{2a})^2 – \frac{b^2}{4a }+c= a(x +\frac{b}{2a})^2+ (c- \frac{b^2}{4a})\)

This is of the form \(a(x+m)^2+n\), where,
\(m= \frac{b}{2a}\), \(n=c – (\frac{b^2}{4a})\)

How to apply completing the square method?

Let’s learn how to apply the completing the square method using an example.

Example: Complete the square in the expression \(-4x^2- 8x-12\).

First, we should make sure that the coefficient of \(x^2\) is \(1\). If the coefficient of \(x^2\) is not \(1\), we will place the number outside as a common factor. We will get:

\(-4x^2- 8x – 12 = -4(x^2 + 2x + 3)\)
Now, the coefficient of \(x^2\) is \(1\).

  • Step 1: Find half of the \(x\)-factor. Here, the coefficient \(x\) is \(2\). Half of \(2\) is \(1\).
  • Step 2: Find the square of the number above. \(1^2=1\)
  • Step 3: Add and subtract the above number after the \(x\) term in the expression whose coefficient of \(x^2\) is \(1\). This means, \(-4(x^2+2x+3) = -4(x^2+2x+1-1+3)\).
  • Step 4: Factorize the perfect square trinomial formed using the first \(3\) terms using the identity \(x^2+2xy+ y^2 = (x + y)^2\). In this case, \(x^2+ 2x+1 = (x + 1)^2\). The above expression from Step \(3\) becomes: \(-4(x^2+ 2x + 1-1+3)= -4((x + 1)^2- 1+3)\)
  • Step 5: Simplify the last two numbers. Here, \(-1+3=2\). Thus, the above expression is: \(-4x^2- 8x – 12 = -4(x + 1)^2-8\). This is of the form \(a(x + m)^2+ n\). Hence, we have completed the square. Thus, \(-4x^2- 8x- 12 = -4(x + 1)^2-8\)

Note: To complete the square in an expression \(ax^2+ bx + c\):

  • Make sure the coefficient of \(x^2\) is \(1\).
  • Add and subtract \((\frac{b}{2})^2\) after the \(x\) term and simplify.

Completing the Square – Example 1:

Use completing the square method to solve \(x^2-4x-5=0\).

Solution:

First, transpose the constant term to the other side of the equation:

\(x^2- 4x = 5\)

Then, take half of the coefficient of the \(x\)-term, which is \(-4\), including the sign, which gives \(-2\). Take the square of \(-2\) to get \(+4\), and add this squared value to both sides of the equation:

\(x^2- 4x+ 4= 5 + 4 ⇒ x^2- 4x + 4 = 9\)
This process creates a quadratic expression that is a perfect square on the left-hand side of the equation. We can replace the quadratic equation with the squared-binomial form:

\((x – 2)^2= 9\)

Now that we have completed the expression to create a perfect-square binomial, let us solve:

\((x-2)^2= 9\)
\((x – 2) = ±\sqrt{9}\)

\(x-2=±3\)
\(x=2+3=5\) , \(x=2-3=-1\)
\(x = 5, -1\)

Exercises for Completing the Square

Solve each equation by completing the square. 

  1. \(\color{blue}{x^2+12x+32=0}\)
  2. \(\color{blue}{x^2-6x-3=0}\)
  3. \(\color{blue}{x^2-10x+16=0}\)
  4. \(\color{blue}{2x^2+7x+6=0}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{x=-4, -8}\)
  2. \(\color{blue}{x=3+2\sqrt{3},\:3-2\sqrt{3}}\)
  3. \(\color{blue}{x=2,8}\)
  4. \(\color{blue}{x=-\frac{3}{2}, -2}\)

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