Question

how do i solve a quadratic function, when leading coefficient equals 1

Answer 1

To solve a quadratic function of the form \( ax^2 + bx + c = 0 \) where the leading coefficient \( a = 1 \), you can simplify the equation to the standard form \( x^2 + bx + c = 0 \). Here’s how to solve it step by step:

### Step 1: Identify the coefficients
From the quadratic equation \( x^2 + bx + c = 0 \), identify:
- \( b \) (the coefficient of \( x \))
- \( c \) (the constant term)

### Step 2: Use the Quadratic Formula
The solutions to the quadratic equation can be found using the quadratic formula:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\]
Since \( a = 1 \) in this case, the formula simplifies to:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4c}}}}{2}
\]

### Step 3: Calculate the Discriminant
The discriminant \( D \) is given by:
\[
D = b^2 - 4c
\]
- If \( D > 0 \): There are two distinct real solutions.
- If \( D = 0 \): There is one real solution (a repeated root).
- If \( D < 0 \): There are no real solutions (the solutions are complex).

### Step 4: Compute the Solutions
1. Calculate \( D \).
2. Depending on the value of \( D \), use the quadratic formula to find the roots:
- If \( D > 0 \): Calculate \
\[
x_1 = \frac{{-b + \sqrt{D}}}{2}, \quad x_2 = \frac{{-b - \sqrt{D}}}{2}
\]
- If \( D = 0 \): Calculate \
\[
x = \frac{{-b}}{2}
\]
- If \( D < 0 \): Indicate that the solutions are complex:
- The roots can be expressed as \
\[
x = \frac{{-b}}{2} \pm \frac{{\sqrt{|D|}}}{2}i
\]

### Example
Let’s say you have the quadratic equation \( x^2 - 5x + 6 = 0 \).

1. Identify coefficients: \( b = -5 \), \( c = 6 \).
2. Calculate the discriminant:
\[
D = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1
\]
3. Since \( D > 0 \), we have two distinct solutions:
\[
x_1 = \frac{{5 + \sqrt{1}}}{2} = \frac{6}{2} = 3
\]
\[
x_2 = \frac{{5 - \sqrt{1}}}{2} = \frac{4}{2} = 2
\]

Thus, the solutions are \( x = 3 \) and \( x = 2 \).