Solve the quadratic equation by factoring. (Enter your answers as a comma-separated list.)
25x^2 + 30x + 9 = 0
note that the first and last terms are perfect squares. So, a good first guess would be
(5x+3)(5x+3) = 0
that works. done.
To solve the quadratic equation 25x^2 + 30x + 9 = 0 by factoring, we need to find two binomial factors such that when multiplied together, they equal the trinomial on the left side of the equation.
Step 1: Look at the coefficient of the x^2 term.
In this case, the coefficient is 25.
Step 2: Find two numbers whose product is equal to the product of the coefficient of the x^2 term and the constant term. In this case, those numbers are 5 and 5 since 5 * 5 = 25.
Step 3: Rewrite the middle term (30x) using the two numbers found in step 2.
We need to rewrite 30x using the numbers 5 and 5. Since the sign of the middle term is positive, we use the numbers with the same sign. Thus, 30x can be rewritten as 5x + 5x, which gives us: 25x^2 + 5x + 5x + 9 = 0.
Step 4: Group the terms.
Rearranging the equation, we get: (25x^2 + 5x) + (5x + 9) = 0.
Step 5: Factor out the common factors from each group.
Factoring out the common factors, we have: 5x(5x + 1) + 9(5x + 1) = 0.
Step 6: Identify the common binomial factor.
The common binomial factor is (5x + 1).
Step 7: Write the factored form of the equation.
Using the common factor, we can factor the equation as follows: (5x + 1)(5x + 9) = 0.
Step 8: Set each factor equal to zero.
Setting (5x + 1) equal to zero, we have: 5x + 1 = 0.
Solving this equation, we find: x = -1/5.
Setting (5x + 9) equal to zero, we have: 5x + 9 = 0.
Solving this equation, we find: x = -9/5.
Therefore, the solutions to the quadratic equation 25x^2 + 30x + 9 = 0, obtained by factoring, are: x = -1/5, -9/5.