**Factor using quadratic pattern :**

To factor the given polynomial equation using quadratic pattern, we need to convert the given polynomial in a quadratic form.

Let us look into some examples to understand the above concept.

**Example 1 :**

Factor using quadratic pattern

x^{4} - 7x^{2} - 18

**Solution :**

= x^{4} - 7x^{2} - 18

= (x^{2})^{2} - 7x^{2} - 18

Let x^{2} = t

= t^{2} - 7t - 18

**Step 1 :**

**By multiplying the coefficient of x ^{2} by the constant term -18, we get -18.**

**Step 2 :**

**Now we need to spit this -18 as two parts, and the product of those parts must be equal to -18 and simplified value must be equal to the middle term (-7).**

**Since the middle and last terms are negative, the factors will be in the combination of positive and negative.**

**-18 = -9 **⋅ **2 and -9 + 2 = -7**

**Step 3 :**

Grouping into linear factors

**= t ^{2} - 9t + 2t - 18**

**= t (t - 9) + 2(t - 9)**

**= (t + 2) (t - 9)**

**Step 4 :**

Replacing "t" by x^{2},

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

**Hence the factors are ****(**x^{2}** + 2) (**x^{2 }- 9**).**

**Example 2 :**

Factor using quadratic pattern

x^{4} - 16x^{2} + 63

**Solution :**

= x^{4} - 16x^{2} + 63

= (x^{2})^{2} - 16x^{2} + 63

Let x^{2} = t

= t^{2} - 16t + 63

**Step 1 :**

**By multiplying the coefficient of x ^{2} by the constant term 63, we get 63.**

**Step 2 :**

**Now we need to spit this 63 as two parts, and the product of those parts must be equal to 63 and simplified value must be equal to the middle term (-16).**

**Since the middle is negative, both factors will have negative sign.**

**63 = -9 **⋅ (-7)** and -9 + (-7) = -16**

**Step 3 :**

Grouping into linear factors

**= t ^{2} - 9t - 7t + 63**

**= t (t - 9) - 7(t - 9)**

**= (t - 9) (t - 7)**

**Step 4 :**

Replacing "t" by x^{2},

**= (**x^{2}** - 9) (**x^{2 }- 7**)**

**Hence the factors are ****(**x^{2}** - 9) (**x^{2 }- 7**).**

**Example 3 :**

Factor using quadratic pattern

7x^{4} - 45x^{2} - 28

**Solution :**

= x^{4} - 45x^{2} - 28

= 7(x^{2})^{2} - 45x^{2} - 28

Let x^{2} = t

= 7t^{2} - 45t - 28

**Step 1 :**

**By multiplying the coefficient of x ^{2} (7) by the constant term 28, we get -196.**

**Step 2 :**

**Now we need to spit this -196 as two parts, and the product of those parts must be equal to -196 and simplified value must be equal to the middle term (-45).**

**Since the middle and last terms are negative, the factors will be in the combination of positive and negative.**

**-196 = -49 **⋅ 4** and -49 + 4 = -45**

**Step 3 :**

Grouping into linear factors

= 7t^{2} - 45t - 28

= 7t^{2} - 49t + 4t - 28

**= 7t (t - 7) + 4(t - 7)**

**= (7t + 4) (t - 7)**

**Step 4 :**

Replacing "t" by x^{2},

**= ****(7**x^{2}** + 4) (**x^{2}** - 7)**

**Hence the factors are ****(7**x^{2}** + 4) (**x^{2}** - 7).**

**Example 4 :**

Factor using quadratic pattern

5x^{4} - x^{2} - 18

**Solution :**

= 5x^{4} - x^{2} - 18

= 5(x^{2})^{2} - x^{2} - 18

Let x^{2} = t

= 5t^{2} - t - 18

**Step 1 :**

**By multiplying the coefficient of x ^{2} (5) by the constant term -18, we get -90.**

**Step 2 :**

**Now we need to spit this -90 as two parts, and the product of those parts must be equal to -90 and simplified value must be equal to the middle term (-1).**

**Since the middle and last terms are negative, the factors will be in the combination of positive and negative.**

**-90 = -10 **⋅ 9** and -10 + 9 = -1**

**Step 3 :**

Grouping into linear factors

= 5t^{2} - t - 18

= 5t^{2} - 10t + 9t - 18

**= 5t (t - 2) + 9(t - 2)**

**= (5t + 9) (t - 2)**

**Step 4 :**

Replacing "t" by x^{2},

**= ****(5**x^{2}** + 9) (**x^{2}** - 2)**

**Hence the factors are **** ****(5**x^{2}** + 9) (**x^{2}** - 2).**

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