Solve the quadratic equation by factoring. Check your solutions in the original equation. (Enter your answers as a comma-separated list.)
x^2 + 2ax + a^2 = 0
(x+a)(x+a)
x = -a or x = -a :)
-a,-a
(-a)^2 + 2a(-a) + a^2
a^2 - 2 a^2 + a^2 = 0 would you believe?
To solve the quadratic equation x^2 + 2ax + a^2 = 0 by factoring, we need to find two numbers whose product is a^2 and whose sum is 2a.
Let's consider the general form of a quadratic equation: ax^2 + bx + c = 0.
In this case, a = 1, b = 2a, and c = a^2.
To factor the quadratic equation, we need to find two numbers, let's call them p and q, such that pq = a^2 and p + q = 2a.
From these conditions, we can determine that p = a and q = a.
So, the factored form of the given quadratic equation is (x + a)(x + a) = 0.
To find the solutions, set each factor equal to zero:
x + a = 0 --> x = -a
Since both factors are the same, we have only one solution: x = -a.
To check the solution, substitute x = -a back into the original equation:
(-a)^2 + 2a(-a) + a^2 = 0
This simplifies to:
a^2 - 2a^2 + a^2 = 0
Simplifying further:
0 = 0
Since the left side equals the right side, we can conclude that x = -a is the correct solution to the given quadratic equation.
Therefore, the solution to the quadratic equation x^2 + 2ax + a^2 = 0 is x = -a.