How can I solve the equation 6x² – 150 by factorization?

To solve the equation 6x² – 150 = 0 by factorization, we’ll follow these steps:

1. Step 1: Simplify the equation.
   Notice that both terms in the equation can be factored by 6, which is a common factor:

6x² – 150 = 0 can be rewritten as 6(x² – 25) = 0.

2. Step 2: Factor the quadratic expression.
   The expression x² – 25 is a difference of squares, which can be factored using the formula a² – b² = (a + b)(a – b). In this case, a is x and b is 5 since 25 is 5². Thus, we have:

x² – 25 = (x + 5)(x – 5).

3. Step 3: Rewrite the whole equation.
   Replace the quadratic expression in the factored form:

6(x + 5)(x – 5) = 0.

4. Step 4: Set each factor to zero.
   To find the values of x, we need to set each factor equal to zero:

6 ≠ 0 (this does not provide a solution),

x + 5 = 0,

x – 5 = 0.

5. Step 5: Solve for x.
   From the factors:
• x + 5 = 0 gives us x = -5.
• x – 5 = 0 gives us x = 5.

Final Solutions:
Therefore, the solutions to the equation 6x² – 150 = 0 are:

x = -5 and x = 5.