Quadratic equations can always be solved using a formula, even when factorisation is difficult or not possible. This method works for all equations of the form
\( x^2 + bx + c = 0 \) and \( ax^2 + bx + c = 0 \).

 

 

The Quadratic Formula

For any quadratic equation written in the form:

$$
ax^2 + bx + c = 0
$$

 

the solutions are given by the quadratic formula:

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

 

The symbol \( \pm \) means there are usually two solutions, one using plus and one using minus.

 

 

Using the Formula for x² + bx + c = 0

When the coefficient of \( x^2 \) is 1, the formula still applies with \( a = 1 \).

 

For example:

$$
x^2 + 6x + 5 = 0
$$

 

Identify the values:

$$
a = 1,\quad b = 6,\quad c = 5
$$

 

Substitute into the formula:

$$
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(5)}}{2(1)}
$$

 

Simplify inside the square root:

$$
x = \frac{-6 \pm \sqrt{36 - 20}}{2}
$$

$$
x = \frac{-6 \pm \sqrt{16}}{2}
$$

$$
x = \frac{-6 \pm 4}{2}
$$

 

This gives two solutions:

$$
x = -1
$$

$$
x = -5
$$

 

 

Using the Formula for ax² + bx + c = 0

When the coefficient of \( x^2 \) is not 1, the same method is used.

 

For example:

$$
2x^2 + 3x - 2 = 0
$$

 

Identify the values:

$$
a = 2,\quad b = 3,\quad c = -2
$$

 

Substitute into the formula:

$$
x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)}
$$

 

Simplify:

$$
x = \frac{-3 \pm \sqrt{9 + 16}}{4}
$$

$$
x = \frac{-3 \pm \sqrt{25}}{4}
$$

$$
x = \frac{-3 \pm 5}{4}
$$

 

This gives:

$$
x = \frac{1}{2}
$$

$$
x = -2
$$

 

 

The Discriminant

The expression under the square root,

$$
b^2 - 4ac
$$

 

is called the discriminant. It tells you how many solutions the equation has.

 

If the discriminant is positive, there are two real solutions.
If it is zero, there is one real solution.
If it is negative, there are no real solutions.

 

 

Key Points to Remember

Always write the equation in the form \( ax^2 + bx + c = 0 \).
Identify \( a \), \( b \) and \( c \) carefully, including signs.
Substitute into the formula accurately.
Simplify step by step to avoid mistakes.

 

The quadratic formula is a reliable method that works for all quadratic equations and is essential when factorisation is not possible.