solve this equation for using the quadratic formula $$ x^{2}=(x-1)^{2}+(x-2)^{2} $$, and $$ x ? $$

Question

UpStudy AI · Accepted Answer

To solve the equation using the **quadratic formula**, follow these steps:

$$
x^{2} = (x - 1)^{2} + (x - 2)^{2}
$$

### Step 1: Expand the Right Side

First, expand the squared terms on the right side of the equation:

$$
(x - 1)^2 = x^2 - 2x + 1
$$
$$
(x - 2)^2 = x^2 - 4x + 4
$$

So, the equation becomes:

$$
x^2 = (x^2 - 2x + 1) + (x^2 - 4x + 4)
$$

### Step 2: Combine Like Terms

Combine the terms on the right side:

$$
x^2 = 2x^2 - 6x + 5
$$

### Step 3: Rearrange the Equation

Move all terms to one side to set the equation to zero:

$$
x^2 - 2x^2 + 6x - 5 = 0
$$
$$
-x^2 + 6x - 5 = 0
$$

Multiply through by -1 to make the coefficient of $$ x^2 $$ positive:

$$
x^2 - 6x + 5 = 0
$$

### Step 4: Apply the Quadratic Formula

The quadratic formula is:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

For the equation $$ x^2 - 6x + 5 = 0 $$:

- $$ a = 1 $$
- $$ b = -6 $$
- $$ c = 5 $$

Plugging these values into the formula:

$$
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}
$$
$$
x = \frac{6 \pm \sqrt{36 - 20}}{2}
$$
$$
x = \frac{6 \pm \sqrt{16}}{2}
$$
$$
x = \frac{6 \pm 4}{2}
$$

### Step 5: Solve for $$ x $$

Calculate both possible solutions:

1. **Positive Root:**
   $$
   x = \frac{6 + 4}{2} = \frac{10}{2} = 5
   $$

2. **Negative Root:**
   $$
   x = \frac{6 - 4}{2} = \frac{2}{2} = 1
   $$

### Step 6: Verify the Solutions

**For $$ x = 1 $$:**
$$
1^2 = (1 - 1)^2 + (1 - 2)^2 \
1 = 0 + 1 \
1 = 1 \quad \checkmark
$$

**For $$ x = 5 $$:**
$$
5^2 = (5 - 1)^2 + (5 - 2)^2 \
25 = 16 + 9 \
25 = 25 \quad \checkmark
$$

Both solutions satisfy the original equation.

### **Final Answer:**

The solutions are $$ x = 1 $$ and $$ x = 5 $$.
The solutions are $$ x = 1 $$ and $$ x = 5 $$.

solve this equation for using the quadratic formula , and

Upstudy AI Solution

Answer

Solution

Step 1: Expand the Right Side

Step 2: Combine Like Terms

Step 3: Rearrange the Equation

Step 4: Apply the Quadratic Formula

Step 5: Solve for

Step 6: Verify the Solutions

Final Answer:

Beyond the Answer

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