Perfect squares

A perfect square is a number that is obtained when a whole number is multiplied by itself. In other words, it is the result of squaring an integer. For example, $\mathrm{1=1\times 1}$ \mathrm{1=1\times 1}$$1=1×1, $\mathrm{4=2\times 2}$ \mathrm{4=2\times 2}$$4=2×2, $\mathrm{9=3\times 3}$ \mathrm{9=3\times 3}$$9=3×3, and $\mathrm{16=4\times 4}$ \mathrm{16=4\times 4}$$16=4×4. These numbers are called perfect squares because they can be arranged to form a perfect square shape.

Perfect squares are important in mathematics because they help us understand square roots, area, and patterns in numbers. The square root of a perfect square is always a whole number. For instance, the square root of 25 is 5, since $\mathrm{5\times 5=25}$ \mathrm{5\times 5=25}$$5×5=25.

in quadratic expressions, a perfect square occurs when the expression can be written as the square of a binomial. This means the quadratic can be expressed in the form:$$(a+b)2\text{or}(a-b)2$$

When expanded, these forms follow special patterns:$$(a + b)^2 = a^2 + 2ab + b^2$$$$(a – b)^2 = a^2 – 2ab + b^2$$

A quadratic expression is called a perfect square trinomial if:

1. The first term is a perfect square.
2. The last term is a perfect square.
3. The middle term is twice the product of the square roots of the first and last terms.

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Examples

x²+6x+9 is a perfect square

x² = (x) x (x) =(x)²

9 = 3 x 3 = (3)²

middle term : 6x = 2(x)(3)

Therefore:

x²+6x+9 = (x+3)²

Example 2

assume x²−10x+25 is a perfect square

x² = (x) (x) = (x)²

25 = (5)²

middle term:-10x = -2(x)(5)

therefore x²−10x+25 = (x-5)²

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Importance in Solving Quadratics

Perfect square expressions are useful because:

• They make factorization easier.
• They help in completing the square, a method used to solve quadratic equations.
• They simplify graphing, since expressions like $(x – a)^2$ clearly show the vertex of a parabola.

Example problem

factorize x² – 8x +16

solution

The constant term is a perfect square

middle term is twice the constant term

hence 16 = (4)(4)

-8 = -4-4

hence x² – 8x +16 = (x-4)(x-4) = (x-4)²

example

factorize:

solution

Example

factorize 4x²+20x+25

solution

4x²= (2x)(2x)

25=(5)(5)

hence 4x²+20x+25 = (2x+5)(2x+5) = (2x+5)²

completing the square

Completing the square is a method used to solve quadratic equations and rewrite quadratic expressions in a special form. It helps us change a quadratic expression into a perfect square trinomial, which can then be factored easily.

consider the expression x² +bx

we can add a constant term such that the expression becomes a perfect square.

To complete the square:

Take half of ,square it and then add it to the expression.

that is: add (b/2)² to the expression

Example problem on completing the square

what must be added to x² -18x to make it a perfect square?

so x² -18x is transformed to x² -18x+81 to be a perfect square.

Related Topics

• Introduction to matrices
• Gradients of line
• Quadratic equation