Extracting Square Roots: Using short multiplication formulas

Let's examine the given equation:

$(x-1)^2=x^2$First, let's simplify the equation, using the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$,

We'll start by opening the parentheses on the left side using the perfect square formula and then move terms and combine like terms, in the final step we'll solve the resulting simplified equation:

$(x-1)^2=x^2 \\ \downarrow\\ x^2-2\cdot x\cdot1+1^2=x^2\\ x^2-2x+1= x^2\\ -2x=-1\hspace{6pt}\text{/}:(-2)\\ \boxed{x=\frac{1}{2}}$Therefore, the correct answer is answer A.

To solve the equation $(-x+1)^2=(2x+1)^2$, we will follow these steps:

• Step 1: Recognize the equation as an application of the identity $a^2 = b^2$. This implies $a = b$ or $a = -b$.
• Step 2: Apply the identity to our equation.
• Step 3: Solve each of the resulting equations individually to find the possible values of $x$.

Now, let's perform each step in detail:

Step 1: We have the equation $(-x+1)^2 = (2x+1)^2$. According to the identity $a^2 = b^2$, we can set up the following cases:
Case 1: $-x + 1 = 2x + 1$,
Case 2: $-x + 1 = -(2x + 1)$.

Step 2: Solve Case 1:
From $-x + 1 = 2x + 1$, subtract 1 from both sides: $-x = 2x$.
Adding $x$ to both sides gives $0 = 3x$.
Divide by 3: $x = 0$.

Step 3: Solve Case 2:
From $-x + 1 = -(2x + 1)$, distribute the negative sign on the right: $-x + 1 = -2x - 1$.
Add $2x$ to both sides: $x + 1 = -1$.
Subtract 1 from both sides: $x = -2$.

Therefore, the solutions to the equation are $x_1=0$ and $x_2=-2$.

The correct answer is:

$x_1=0,x_2=-2$

Let's examine the given equation:

$(x+2)^2-12=x^2$First, let's simplify the equation, for this we'll make use of the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$,

We'll start by opening the parentheses on the left side using the perfect square formula and then move terms and combine like terms, in the final step we'll solve the simplified equation we obtain:

$(x+2)^2-12=x^2 \\ \downarrow\\ x^2+2\cdot x\cdot2+2^2-12=x^2\\ x^2+4x+4-12= x^2\\ 4x=8\hspace{6pt}\text{/}:4\\ \boxed{x=2}$Therefore, the correct answer is answer C.

Let's examine the given equation:

$(x+1)^2=x^2$First, let's simplify the equation, using the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$,

We'll start by opening the parentheses on the left side using the perfect square formula and then move terms and combine like terms, in the final step we'll solve the simplified equation we obtain:

$(x+1)^2=x^2 \\ \downarrow\\ x^2+2\cdot x\cdot1+1^2=x^2\\ x^2+2x+1= x^2\\ 2x=-1\hspace{6pt}\text{/}:2\\ \boxed{x=-\frac{1}{2}}$Therefore, the correct answer is answer A.

Let's examine the given equation:

$(x+3)^2=(x-3)^2$First, let's simplify the equation, for this we'll use the perfect square formula for a binomial squared:

$(a\pm b)^2=a^2\pm2ab+b^2$,

We'll start by opening the parentheses on both sides simultaneously using the perfect square formula mentioned, then we'll move terms and combine like terms, and in the final step we'll solve the resulting simplified equation:

$(x+3)^2=(x-3)^2 \\ \downarrow\\ x^2+2\cdot x\cdot3+3^2= x^2-2\cdot x\cdot3+3^2 \\ x^2+6x+9= x^2-6x+9 \\ x^2+6x+9- x^2+6x-9 =0\\ 12x=0\hspace{6pt}\text{/}:12\\ \boxed{x=0}$Therefore, the correct answer is answer A.