The Quadratic Formula

What is the Quadratic Formula?

The quadratic formula is a tool used to solve any quadratic equation of the form: $$ax^2 + bx + c = 0$$

It works even when factorising or completing the square seems tricky. The formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

 

Breaking It Down

1. $a$: The coefficient of $x^2$
2. $b$: The coefficient of $x$
3. $c$: The constant (number without $x$).
4. $\pm$: This means there are two solutions, one for $+$ and one for $-$

 

Understanding the Discriminant

The discriminant, $b^2 - 4ac$, is key to predicting the nature of the roots:

• If $b^2 - 4ac > 0$, the equation has two distinct real roots.
• If $b^2 - 4ac = 0$, the equation has one real root (roots are equal).
• If $b^2 - 4ac < 0$, the equation has two complex roots.

 

Steps to Solve Using the Quadratic Formula

1. Write the equation in standard form: $$ax^2 + bx + c = 0$$

2. Identify $a$, $b$ and $c$ from the equation.
3. Substitute $a$, $b$ and $c$ into the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
4. Simplify the discriminant $(b^2 - 4ac)$.
5. Calculate the two solutions by using $+$ and $-$ for the $\pm$ symbol.
6. Simplify the final answers.

 

Examples

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

1. Identify coefficients: $$a = 1, b = 6, c = 5$$
2. Substitute into the formula: $$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(5)}}{2(1)}$$
   
3. Simplify the discriminant: $$x = \frac{-6 \pm \sqrt{36 - 20}}{2}$$
   
4. Simplify further: $$x = \frac{-6 \pm \sqrt{16}}{2}$$
   

5. Find the two solutions:

   • For $+$: $$x = \frac{-6 + 4}{2} = -1$$
   • For $-$: $$x = \frac{-6 - 4}{2} = -5$$

Solution: $$x = -1 \quad \text{or} \quad x = -5$$