**Solve a quadratic equation by factoring :**

Usually we have three methods to solve a quadratic equations.

(1) Factoring

(2) Using quadratic formula

(3) Completing the square method

Here we are going to see how to solve a quadratic equation by using the method factoring.

To factor a quadratic equation which is in the form ax² + bx + c = 0, we have to follow the steps given below.

- If it is 1, then we have to take the constant term and split it into two factors.
- In which the product of two factors must be equal to the constant term and the simplified value of those factors equal to the middle term, that is coefficient of x.
- Write each factors in the form (x + a) (x + b) according to the sign.
- Set each factors equal to zero to get the value of x.

- Multiply the coefficient of x² and constant term.
- Split this product into two factors such that their sum is equal to the coefficient of x .
- Divide each factors by the coefficient of x².
- If it is possible we may simplify, otherwise we have to write the denominator along with x.
- write each factors in the form (x + a) (x + b).
- Set each factors equal to zero to get the value of x.

**Example 1 :**

Solve x² + 9 x + 14 = 0

**Solution :**

**Since the coefficient of x**² is 1, split the constant term that into two parts.

14 = 2 ⋅ 7, 2 + 7 = 9

x² + 9 x + 14 = 0

(x + 2) (x + 7) = 0

x - 2 = 0 x + 7 = 0

x = 2, x = -7

Hence the solution is {2, -7}

**Example**** 2 :**

Solve x² - 9 x + 14 = 0

**Solution :**

**Since the coefficient of x**² is 1, split the constant term that into two parts.

14 = -2 ⋅ (-7) , -2 - 7 = -9

x² - 9 x + 14 = (x - 2) (x - 7)

(x - 2) (x - 7) = 0

x - 2 = 0 x - 7 = 0

x = 2, x = 7

Hence the solution is {2, 7}

**Example 3 :**

Solve x² + 2 x - 15 = 0

**Solution :**

**Since the coefficient of x**² is 1, split the constant term that into two parts.

-15 = -3 ⋅ 5 , -3 + 5 = 2

x² + 2 x - 15 = (x - 3) (x + 5)

(x - 3) (x + 5) = 0

x - 3 = 0 x + 5 = 0

x = 3, x = -5

Hence the solution is {3, -5}

**Example 4 :**

Solve x² - 2 x - 15 = 0

**Solution :**

**Since the coefficient of x**² is 1, split the constant term that into two parts.

-15 = -5 ⋅ 3 , -5 + 3 = -2

x² - 2 x - 15 = (x - 5) (x + 3)

(x - 5) (x + 3) = 0

x - 5 = 0 x + 3 = 0

x = 5, x = -3

Hence the solution is {-3, 5}

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