Find all real zeros of f (x) = 2x^2 + 5x-18 algebraically.

The zeros are {-4.5, 2}.

Step-by-step explanation:

2x^2 + 5x - 18 = 0

Use the 'ac' method of solution.

2 * - 18 = - 36; we need 2 numbers whose product is - 36 and whose sum is + 5.

+9 and - 4 look good so:

2x^2 + 5x - 18 = 0 Replacing the + 5x by - 4x + 9x:

2x^2 - 4x + 9x - 18 = 0

2x (x - 2) + 9 (x - 2) = 0

x - 2 is common to both parts, so:

(2x + 9) (x - 2) = 0

2x + 9 = 0 gives x = - 4.5 and

x - 2 = 0 gives x = 2.

2x2-5x-18=0

Two solutions were found:

x = - 2

x = 9/2 = 4.500

Step by step solution:

Step 1:

Equation at the end of step 1:

(2x2 - 5x) - 18 = 0

Step 2:

Trying to factor by splitting the middle term

2.1 Factoring 2x2-5x-18

The first term is, 2x2 its coefficient is 2.

The middle term is, - 5x its coefficient is - 5.

The last term, "the constant", is - 18

Step-1 : Multiply the coefficient of the first term by the constant 2 • - 18 = - 36

Step-2 : Find two factors of - 36 whose sum equals the coefficient of the middle term, which is - 5.

-36 + 1 = - 35

-18 + 2 = - 16

-12 + 3 = - 9

-9 + 4 = - 5 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, - 9 and 4

2x2 - 9x + 4x - 18

Step-4 : Add up the first 2 terms, pulling out like factors:

x • (2x-9)

Add up the last 2 terms, pulling out common factors:

2 • (2x-9)

Step-5 : Add up the four terms of step 4:

(x+2) • (2x-9)

Which is the desired factorization

Equation at the end of step 2:

(2x - 9) • (x + 2) = 0

Step 3:

Theory - Roots of a product:

3.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.