**Standard form to vertex form by completing the square :**

Here we are going to see how to convert quadratic equation from standard form to vertex form.

**(y - k) = a(x - h) ^{2}**

(i) First we have to check whether the coefficient of x² is 1 or not. If yes we can follow the second step. Otherwise factor out the coefficient of x² from the equation.

(ii) Keep the x^{2} and x terms in the right side and keep the y-term and constant term to the left side.

(iii) Add the square of half of the coefficient of "x" on both sides.

(iv) Now the three terms on the right side will be in the form of a² + 2 a b + b² (or) a² - 2 ab + b².

a² + 2 a b + b² = (a + b)^{2}

a² - 2 a b + b² = (a - b)^{2}

(v) Now we can get the vertex from this equation.

Vertex of the parabola V (h, k)

**Example 1 :**

Use the information provided to write the vertex form equation of each parabola and sketch the parabola.

y = x^{2} - 4x + 3

**Solution :**

**Step 1 :**

**The coefficient of x ^{2} is 1**

**Step 2 :**

**Subtract 3 on both sides**

**y - 3 = x ^{2} - 4x + 3 - 3**

**y - 3 = ****x ^{2} - 4x**

**Step 3 :**

**Half of the of coefficient of x is 2**

**Square of half of the coefficient of x is 2 ^{2} = 4**

**y - 3 + 4 = ****x ^{2} - 4x + 4**

y - 3 + 4 = x^{2} - 2⋅x⋅2 + 2^{2}

y + 1 = (x - 2)^{2}

**Step 4 :**

**By comparing the above equation with vertex form**

(y - k) = a(x - h)^{2}

we can get the vertex.

(h, k) ==> (2, -1)

**Graph of the given parabola :**

Since the coefficient of x^{2} is positive the parabola opens upward.

**Example 2 :**

Use the information provided to write the vertex form equation of each parabola and sketch the parabola.

y = 2x^{2} - 8x + 9

**Solution :**

**Step 1 :**

**The coefficient of x ^{2} is not 1, we have to divide the equation on the right side by 2.**

**y - 9 = 2x ^{2} - 8x **

**y - 9 = 2(x ^{2} - 4x)**

**Step 2 :**

**y - 9 = 2(x ^{2} - 2 x 2 + 2^{2} - 2^{2})**

**y - 9 = 2[(x - 2) ^{2} - 4]**

**y - 9 = 2(x - 2) ^{2} - 8**

**y - 9 + 8 = 2(x - 2) ^{2} - 8 + 8**

**y - 1 = 2 (x - 2) ^{2}**

**Vertex of the parabola is (2, 1)**

After having gone through the stuff given above, we hope that the students would have understood "Standard form to vertex form by completing the square".

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