Question

Quadratic formulae

Answer 1

The quadratic formula is used to find the solutions (roots) of a quadratic equation, which is typically in the form:

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

where:
- \( a \), \( b \), and \( c \) are coefficients,
- \( a \neq 0 \) (if \( a = 0 \), the equation is linear, not quadratic).

The quadratic formula is given by:

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

In this formula:
- \( x \) represents the variable for which you are solving the equation,
- \( \pm \) indicates that there are generally two solutions: one obtained by using the plus sign and the other by using the minus sign,
- The term \( b^2 - 4ac \) is called the discriminant. It helps determine the nature of the roots:
- If \( b^2 - 4ac > 0 \): There are two distinct real roots.
- If \( b^2 - 4ac = 0 \): There is exactly one real root (a repeated root).
- If \( b^2 - 4ac < 0 \): There are two complex (imaginary) roots.

### Example:
For the equation \( 2x^2 - 4x - 6 = 0 \):
- \( a = 2 \)
- \( b = -4 \)
- \( c = -6 \)

Plugging these values into the quadratic formula:

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

2. Substitute into the formula:
\[
x = \frac{{4 \pm \sqrt{64}}}{{2 \cdot 2}} = \frac{{4 \pm 8}}{4}
\]

3. Calculate the two possible values for \( x \):
\[
x_1 = \frac{{4 + 8}}{4} = \frac{12}{4} = 3
\]
\[
x_2 = \frac{{4 - 8}}{4} = \frac{-4}{4} = -1
\]

So the solutions are \( x = 3 \) and \( x = -1 \).