**Factoring quadratics :**

Here we are going to see how to factor quadratic equation.A equation which is in the form of ax² + bx + c is known as quadratic equation. Here a,b and c are just numbers.

To factor a quadratic equation, we have to follow the steps given below.

**Step 1 :**

Multiply the coefficient of x^{2} by the constant term

**Step 2 :**

Split this number into two parts and the product of those parts must be equal to this number and simplified value must be equal to the middle term.

**Step 3 :**

Group them into linear factor

Let us look into some example problems to understand the above concept.

**Example 1 :**

Factor x² + 7 x + 12

**Solution :**

**Step 1 :**

**By multiplying the coefficient of x ^{2} by the constant term 12, we get 12.**

**Step 2 :**

**Now we need to spit this 12 as two parts, and the product of those parts must be equal to 12 and simplified value must be equal to the middle term (7).**

= x^{2} + 3x + 4x + 12

**Step 3 :**

Grouping into linear factors

= x (x + 3) + 4 (x + 3)

= (x + 4) (x + 3)

**Example 2 :**

Factor y² - 16 y + 60

**Solution :**

**Step 1 :**

**By multiplying the coefficient of y ^{2} by the constant term 60, we get 60.**

**Step 2 :**

**Now we need to spit this 60 as two parts, and the product of those parts must be equal to 60 and simplified value must be equal to the middle term (-16).**

**Since the middle term is negative, both factors will be negative.**

= y^{2} - 10y - 6y + 60

**Step 3 :**

Grouping into linear factors

= y (y - 10) - 6 (y - 10)

= (y - 10) (y - 6)

**Example 3 :**

Factor x² + 9x - 22

**Solution :**

**Step 1 :**

**By multiplying the coefficient of x ^{2} by the constant term -22, we get -22.**

**Step 2 :**

**Now we need to spit this -22 as two parts, and the product of those parts must be equal to -22 and simplified value must be equal to the middle term (9).**

**Since the last term is negative, the factors will be in the combination of positive and negative.**

= x^{2} + 11x - 2x - 22

**Step 3 :**

Grouping into linear factors

= x (x + 11) - 2 (x + 11)

= (x + 11) (x - 2)

**Example 4 :**

Factor x² - 2x - 99

**Solution :**

**Step 1 :**

**By multiplying the coefficient of x ^{2} by the constant term -99, we get -99.**

**Step 2 :**

**Now we need to spit this -99 as two parts, and the product of those parts must be equal to -99 and simplified value must be equal to the middle term (-2).**

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

= x^{2} - 11x + 9x - 99

**Step 3 :**

Grouping into linear factors

= x (x - 11) + 9 (x - 11)

= (x - 11) (x + 9)

**Example 5 :**

Factor 3x² + 19x + 6

**Solution :**

**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 (19).**

= 3x^{2} + 1x + 18x + 6

**Step 3 :**

Grouping into linear factors

= x (3x + 1) + 6 (3x + 1)

= (x + 6) (3x + 1)

**Example 6 :**

Factor 9x² - 16x + 7

**Solution :**

**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 term is negative, both factors will be negative.**

= 9x^{2} - 9x - 7x + 7

**Step 3 :**

Grouping into linear factors

= 9x (x - 1) - 7 (x - 1)

= (x - 1) (9x - 7)

**Example 7 :**

Factor 2x² + 17x - 30

**Solution :**

**Step 1 :**

**By multiplying the coefficient of x ^{2} by the constant term -30, we get -30.**

**Step 2 :**

**Now we need to spit this -30 as two parts, and the product of those parts must be equal to -30 and simplified value must be equal to the middle term (17).**

**Since the last term is negative, the factors will be in the combination of positive and negative.**

= 2x^{2} + 20x - 3x - 30

**Step 3 :**

Grouping into linear factors

= 2x (x + 10) - 3 (x + 10)

= (x + 10) (2x - 3)

Let us see the next example on "Factoring quadratics"

**Example 8 :**

Factor 18x² - x - 4

**Solution :**

**Step 1 :**

**By multiplying the coefficient of x ^{2} by the constant term -72, we get -72.**

**Step 2 :**

**Now we need to spit this -72 as two parts, and the product of those parts must be equal to -72 and simplified value must be equal to the middle term (-1).**

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

= 18x^{2} - 9x + 8x - 4

**Step 3 :**

Grouping into linear factors

= 9x (2x - 1) + 4 (2x - 1)

= (2x - 1) (9x + 4)

- Factoring quadratic equations when the coefficient of x² is not 1
- Factoring using Algebraic identities
- Solving quadratic equation using sompleting the square method
- Solving quadratic equation by using quadratic formula
- Practical problem using quadratic equations

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