# 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. 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.

## 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. $$\color{blue}{2x^2+7x+6=0}$$
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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