5x = x²+14 answer using quadratic formula

To solve the equation \( 5x = x^2 + 14 \) using the quadratic formula, follow these steps:

### Step 1: Rearrange the equation into standard quadratic form.
First, bring all terms to one side of the equation:

\[
x^2 - 5x + 14 = 0
\]

### Step 2: Identify coefficients.
The quadratic formula is given by:

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

For the equation \( x^2 - 5x + 14 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -5 \)
- \( c = 14 \)

### Step 3: Apply the quadratic formula.
Substitute the values of \( a \), \( b \), and \( c \) into the formula:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(14)}}{2(1)}
\]

Simplify the expression:

\[
x = \frac{5 \pm \sqrt{25 - 56}}{2}
\]

\[
x = \frac{5 \pm \sqrt{-31}}{2}
\]

### Step 4: Simplify the result.
Since \( \sqrt{-31} \) is imaginary, we write the solution in terms of imaginary numbers:

\[
x = \frac{5 \pm \sqrt{31}i}{2}
\]

Thus, the two solutions are:

\[
x = \frac{5 + \sqrt{31}i}{2} \quad \text{or} \quad x = \frac{5 - \sqrt{31}i}{2}
\]

These are the complex solutions to the equation.

Would you like further details or have any questions?

### Related Questions:
1. What is the general form of the quadratic equation?
2. How do you interpret complex numbers geometrically?
3. Can quadratic equations have real solutions if the discriminant is negative?
4. How does the quadratic formula change when the discriminant is zero?
5. How do you simplify square roots of negative numbers?

### Tip:
When solving quadratic equations, always check the discriminant \( b^2 - 4ac \) to determine the nature of the solutions (real or complex).

5x = x² + 14 answer using quadratic formula

Learn how to solve the quadratic equation 5x = x² + 14 using the quadratic formula. This problem involves key algebraic concepts such as quadratic equations, complex numbers, and the quadratic formula theorem. Suitable for students in Grades 9-12.

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