A quadratic equation ax2 + bx + c = 0, such that a + b + c = 0, has equal roots. Which of the following statements must be true?b = 2aa = 4ca2 + b2 = c2a = c

To determine which of the given statements must be true for the quadratic equation $ax^2 + bx + c = 0$ with the condition $a + b + c = 0$ and equal roots, let's analyze the problem step by step.

Step 1: Understanding Equal Roots

For a quadratic equation $ax^2 + bx + c = 0$ to have equal roots, the discriminant must be zero. The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by: $\Delta = b^2 - 4ac$ For the roots to be equal: $\Delta = 0$ $b^2 - 4ac = 0$ $b^2 = 4ac$

Step 2: Using the Given Condition

We are given that: $a + b + c = 0$ Rearranging this equation, we get: $c = -a - b$

Step 3: Substituting $c$ in the Discriminant Condition

Substitute $c = -a - b$ into the discriminant condition $b^2 = 4ac$: $b^2 = 4a(-a - b)$ $b^2 = -4a^2 - 4ab$ $b^2 + 4ab + 4a^2 = 0$ $(b + 2a)^2 = 0$ Taking the square root of both sides: $b + 2a = 0$ $b = -2a$

Step 4: Verifying the Statements

Now, let's verify each of the given statements:

1. $b = 2a$:

   • From our derived condition $b = -2a$, this statement is false.

2. $a = 4c$:

   • Substitute $c = -a - b$ and $b = -2a$: $c = -a - (-2a)$ $c = -a + 2a$ $c = a$
   • Therefore, $a = c$, not $a = 4c$. This statement is false.

3. $a^2 + b^2 = c^2$:

   • Substitute $b = -2a$ and $c = a$: $a^2 + (-2a)^2 = a^2$ $a^2 + 4a^2 = a^2$ $5a^2 = a^2$
     • This is not true for any non-zero $a$. This statement is false.

4. $a = c$:

   • From our substitution, we found $c = a$. This statement is true.

Conclusion

The only statement that must be true is: $a = c$