Find the root of the equation (5x + 7)^2 = (5x - 6)^2, and give it as a decimal.

Step-by-step solution

1. 1

   Use the equal-squares principle instead of expanding. For real numbers, $A^{2}=B^{2}$ holds exactly when $A=B$ or $A=-B$. Expanding both sides would also work — the $25x^{2}$ terms cancel — but the case split is faster and shows the structure.

2. 2

   Case 1: $5x+7=5x-6$. Subtracting $5x$ from both sides leaves

   $7=-6$

   a false statement with no $x$ in it, so this case contributes no solutions. That is expected: the two linear expressions have the same slope, so they are never equal.

3. 3

   Case 2: $5x+7=-(5x-6)$. Distribute the minus sign on the right:

   $5x+7=-5x+6$

   Collect the $x$-terms on the left and constants on the right:

   $10x=-1$

4. 4

   Solve and convert to a decimal.

   $x=-\frac{1}{10}=-0.1$

5. 5

   Check in the original equation. With $x=-0.1$: $5(-0.1)+7=6.5$ and $5(-0.1)-6=-6.5$. Then $6.5^{2}=42.25$ and $(-6.5)^{2}=42.25\ \checkmark$ — the two brackets are exact opposites, which is precisely what case 2 was engineered to produce.