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
- Calculate \( D \).
- 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
\[ x = \frac{{-b}}{2} \pm \frac{{\sqrt{|D|}}}{2}i \]
- The roots can be expressed as
- If \( D > 0 \): Calculate
Example
Let’s say you have the quadratic equation \( x^2 - 5x + 6 = 0 \).
- Identify coefficients: \( b = -5 \), \( c = 6 \).
- Calculate the discriminant: \[ D = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1 \]
- 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 \( x = 2 \).