Solve each of the following equatins :
`x^(2)+5=(9)/(2)x`

Answer

To solve the quadratic equation \( x^2 + 5 = \frac{9}{2}x \), we will follow these steps:

### Step 1: Rearrange the equation
First, we need to move all terms to one side of the equation. We can do this by subtracting \(\frac{9}{2}x\) from both sides:

\[ x^2 - \frac{9}{2}x + 5 = 0 \]

### Step 2: Eliminate the fraction
To eliminate the fraction, we can multiply the entire equation by 2:

\[ 2(x^2) - 2\left(\frac{9}{2}x\right) + 2(5) = 0 \]

This simplifies to:

\[ 2x^2 - 9x + 10 = 0 \]

### Step 3: Factor the quadratic equation
Next, we will factor the quadratic equation \(2x^2 - 9x + 10 = 0\). We need to find two numbers that multiply to \(2 \times 10 = 20\) and add up to \(-9\). The numbers that satisfy this are \(-4\) and \(-5\).

We can rewrite the middle term:

\[ 2x^2 - 4x - 5x + 10 = 0 \]

### Step 4: Group the terms
Now, we will group the terms:

\[ (2x^2 - 4x) + (-5x + 10) = 0 \]

### Step 5: Factor by grouping
Now we can factor each group:

\[ 2x(x - 2) - 5(x - 2) = 0 \]

### Step 6: Factor out the common term
Now we can factor out the common term \((x - 2)\):

\[ (x - 2)(2x - 5) = 0 \]

### Step 7: Solve for \(x\)
Now we can set each factor to zero:

1. \(x - 2 = 0\) gives us \(x = 2\)
2. \(2x - 5 = 0\) gives us \(2x = 5\) or \(x = \frac{5}{2}\)

### Final Solutions
The solutions to the equation \(x^2 + 5 = \frac{9}{2}x\) are:

\[ x = 2 \quad \text{and} \quad x = \frac{5}{2} \]

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