The number of the real roots of the equation `(x + 1)^2 + |x – 5|=(27)/(4)` is _______ .

Answer

To find the number of real roots of the equation \((x + 1)^2 + |x - 5| = \frac{27}{4}\), we will analyze the equation step by step.

### Step 1: Rewrite the equation
We start with the equation:
\[ (x + 1)^2 + |x - 5| = \frac{27}{4} \]

### Step 2: Analyze the absolute value
The absolute value \(|x - 5|\) can be split into two cases based on the value of \(x\):
1. Case 1: \(x \geq 5\) → \(|x - 5| = x - 5\)
2. Case 2: \(x < 5\) → \(|x - 5| = 5 - x\)

### Step 3: Solve Case 1 (\(x \geq 5\))
In this case, we substitute \(|x - 5|\) in the original equation:
\[ (x + 1)^2 + (x - 5) = \frac{27}{4} \]
This simplifies to:
\[ (x + 1)^2 + x - 5 = \frac{27}{4} \]
\[ (x + 1)^2 + x - 5 - \frac{27}{4} = 0 \]
To eliminate the fraction, multiply through by 4:
\[ 4(x + 1)^2 + 4x - 20 - 27 = 0 \]
\[ 4(x + 1)^2 + 4x - 47 = 0 \]

### Step 4: Expand and simplify
Expanding \(4(x + 1)^2\):
\[ 4(x^2 + 2x + 1) + 4x - 47 = 0 \]
\[ 4x^2 + 8x + 4 + 4x - 47 = 0 \]
\[ 4x^2 + 12x - 43 = 0 \]

### Step 5: Use the quadratic formula
Now we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 4\), \(b = 12\), and \(c = -43\).
\[ D = b^2 - 4ac = 12^2 - 4 \cdot 4 \cdot (-43) = 144 + 688 = 832 \]
Since \(D > 0\), there are 2 real roots.

### Step 6: Solve Case 2 (\(x < 5\))
Now we consider the second case:
\[ (x + 1)^2 + (5 - x) = \frac{27}{4} \]
This simplifies to:
\[ (x + 1)^2 + 5 - x = \frac{27}{4} \]
\[ (x + 1)^2 - x + 5 - \frac{27}{4} = 0 \]
Multiply through by 4 to eliminate the fraction:
\[ 4(x + 1)^2 - 4x + 20 - 27 = 0 \]
\[ 4(x + 1)^2 - 4x - 7 = 0 \]

### Step 7: Expand and simplify
Expanding \(4(x + 1)^2\):
\[ 4(x^2 + 2x + 1) - 4x - 7 = 0 \]
\[ 4x^2 + 8x + 4 - 4x - 7 = 0 \]
\[ 4x^2 + 4x - 3 = 0 \]

### Step 8: Use the quadratic formula again
Now we apply the quadratic formula again:
Here, \(a = 4\), \(b = 4\), and \(c = -3\).
\[ D = b^2 - 4ac = 4^2 - 4 \cdot 4 \cdot (-3) = 16 + 48 = 64 \]
Since \(D > 0\), there are 2 real roots.

### Step 9: Count the total number of real roots
From both cases, we have:
- 2 real roots from Case 1 (\(x \geq 5\))
- 2 real roots from Case 2 (\(x < 5\))

Thus, the total number of real roots of the equation is:
\[ \text{Total real roots} = 2 + 2 = 4 \]

### Final Answer
The number of real roots of the equation is **4**.

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