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When a quadratic expression of the form ax.x + bx + c is converted to vertex form (x – h) (x – h) + k, completing the square formula is required. Solving a quadratic equation is one of the most common applications of completing the square. This formula of mathematics looks very tough. Isn’t it? But, let me tell you this is one of the easiest formulas to solve a quadratic equation if solved step-wise properly. In this article, we shall look at where the completing the square formula is used, the formula given for completing the square, and some examples regarding this formula.

**Some Important Areas Where Completing the Square Method is Used**

The following areas where completing the square is mostly used are:

- Completing the square formula is used to convert the quadratic form into the vertex form.
- To analyse at which point the maximum or minimum value of a quadratic equation is involved.
- Completing the square formula is used to solve a quadratic equation, to graph a quadratic function, and to derive a quadratic formula.

**Formula for Completing the Square**

- The formula given for completing the square is: ax.x + bx + c ⇒ a (x + m) (x + m) + n

where, m is defined as a real number and n is defined as a constant term.

- For completing the square formula, instead of using a complex process, we can use a simple formula to complete the square. To complete the square in the expression ax.x + bx + c, first find:

m = b/2a and n = c – (b.b /4a)

- Substitute these values in the square formula that is: ax2 + bx + c = a (x + m) (x + m) + n. These formulas are derived from geometry. We shall see some examples regarding completing the square formula so that this topic becomes clearer to you.

**Some Examples of Completing the Square Formula**

We will learn to solve a problem by completing the square formula. Let us find the number that should be added to x.x – 7x so that it makes a perfect square trinomial?

Answer:

- The expression given is x.x – 7x.
- Comparing the given expression with ax.x + bx + c, we get a = 1; b = -7
- Using the completing square formula, the term that should be added to make the given expression a perfect square trinomial is,

(b/2a) (b/2a) = (-7/2 (1)) (-7/2 (1) ) = 49/4.

- Therefore, by using the completing square method the number that should be added to the given equation to make it a perfect square trinomial is, 49/4.

**Let us see another example.**

We will use completing the square formula to solve: x.x – 4x – 8 = 0.

Solution:

- Using formula, ax.x + bx + c = a (x + m) (x +m) + n. Where, a = 1, b = -4, c = -8

⇒m = b/2 = (-4)/2(1)= -2

And, n = c – (b.b/4a)= -8 – (-4)(-4)/4(1) = -12

⇒ x2 – 4x – 8= (x-2)(x-2) -12.

⇒ (x-2)2 = 12

⇒ (x-2) = ± √12

⇒ x-2 = ± 2√3

⇒ x = 2 ± 2√3

Thus the value of x using completing the square formula is = 2 ± 2√3. To learn more about these interesting concepts in detail and in an interactive and fun way visit cuemath.com.

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