Factor and solve the following equation 2x^2 + x - 21 = 0.

The factors are (2x + 7)(x - 3) and the solutions are -3.5 and 3

Step-by-step explanation:

* Lets explain how to factor a trinomial in the form ax² ± bx ± c:

- Look at the c term first.  

# If the c term is a positive number, then its factors r , s  will both

  be positive or both be negative.

# a has two factors h and k

# The sum of c and a is b.

# The brackets are (hx ± r)(kx ± s) where a = hk , c = rs and b = rk + hs

# If the c term is a negative number, then either r or s will be negative,

   but not both.

# a has two factors h and k

# The difference of c and a is b.

# The brackets are (hx + r)(kx - s) where a = hk , c = rs and b = rk - hs

* Lets solve the problem

∵ The equation is 2x² + x - 21 = 0

∵ The general form of the equation is ax² + bx + c = 0

∴ a = 2 , b = 1 , c = -21

∵ c is negative

∴ its factors r and s have different sign

∵ a = 2

∵ The factors of a are h , k

∵ 2 = 2 × 1

∴ h = 2 and k = 1

∵ -21 = 7 × -3

∴ r = 7 and s = -3

∵ The brackets are (hx + r)(kx - s)

∴ 2x² + x - 21 = (2x + 7)(x - 3)

∵ 2x² + x - 21 = 0

∴ (2x + 7)(x - 3) = 0

- Equate each bracket by 0

∴ 2x + 7 = 0 ⇒ subtract 7 from both sides

∴ 2x = -7 ⇒ divide both sides by 2

∴ x = -7/2 = -3.5

- OR

∴ x - 3 = 0 ⇒ add 3 to both sides

∴ x = 3

∴ The solutions are -3.5 and 3

* The factors are (2x + 7)(x - 3) and the solutions are -3.5 and 3