Final answer:
To solve the equation x^2+14x+49=49, we factor the left hand side to (x+7)^2, then take the square root of both sides, which results in two real solutions, x = 0 and x = -14.
Step-by-step explanation:
Finding the Real Solutions of a Quadratic Equation by Factoring
To find all real solutions of the equation x^2+14x+49=49, we start by factoring the left hand side. Notice that the left side is a perfect square trinomial, which factors into (x+7)^2. Our equation becomes:
(x+7)^2 = 49
Since the right hand side of the equation is already a perfect square (49 is 7^2), we can take the square root of both sides, yielding:
x+7 = ±7
Now, we can solve for two possible values of x:
x+7 = 7 or x+7 = -7
x = 0 x = -14
These are the two real solutions to the equation. By factoring the left hand side and taking a square root, we could efficiently solve for x without needing the quadratic formula.