Question

quadratic equation roots solution

Answer 1

A quadratic equation is typically expressed in the standard form:

\[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are coefficients, and \( a \neq 0 \).

To find the roots of the quadratic equation, you can use the quadratic formula:

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

Here’s a breakdown of the steps:

1. **Identify the coefficients**: Determine the values of \( a \), \( b \), and \( c \) from the quadratic equation.
2. **Calculate the discriminant**: Compute \( D = b^2 - 4ac \).
- If \( D > 0 \): There are two distinct real roots.
- If \( D = 0 \): There is one real root (a repeated root).
- If \( D < 0 \): There are two complex (conjugate) roots.
3. **Use the quadratic formula**: Plug in the values of \( a \), \( b \), and \( D \) into the quadratic formula to find the roots.

### Example

Let's solve the quadratic equation \( 2x^2 - 4x - 6 = 0 \).

1. **Identify coefficients**:
- \( a = 2 \)
- \( b = -4 \)
- \( c = -6 \)

2. **Calculate the discriminant**:
\[
D = b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64
\]

3. **Use the quadratic formula**:
\[
x = \frac{{-(-4) \pm \sqrt{64}}}{{2 \times 2}} = \frac{{4 \pm 8}}{{4}}
\]

This gives us two solutions:
\[
x_1 = \frac{12}{4} = 3
\]
\[
x_2 = \frac{-4}{4} = -1
\]

Thus, the roots of the equation \( 2x^2 - 4x - 6 = 0 \) are \( x = 3 \) and \( x = -1 \).