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

Answer

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

### Step 1: Rearrange the equation
First, we need to rearrange the equation to bring all terms to one side, resulting in a standard form of a quadratic equation.

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

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

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

This simplifies to:

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

### Step 3: Factor the quadratic equation
Next, we need to factor the quadratic equation \( 2x^2 - 9x + 10 = 0 \). We are looking for two numbers that multiply to \( 2 \cdot 10 = 20 \) and add up to \( -9 \).

The numbers that satisfy these conditions are \( -4 \) and \( -5 \).

### Step 4: Rewrite the middle term
Now, we can rewrite the equation using these numbers:

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

### Step 5: Factor by grouping
Next, we will group the terms:

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

Factoring out the common factors from each group gives us:

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

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

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

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

1. \( 2x - 5 = 0 \)
\[ 2x = 5 \implies x = \frac{5}{2} \]

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

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

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

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