1.03 Solve using the quadratic formula
There are four ways to solve a quadratic equation (i.e. an equation of the form $ax^2+bx+c=0$ax2+bx+c=0):
- By algebraic manipulation
- By factoring and using the null factor law
- By completing the square
- By using the quadratic formula
If $ax^2+bx+c=0$ax2+bx+c=0, then:
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$x=−b±√b2−4ac2a
The advantage of using the quadratic formula is that it always works (unlike factoring) and it always follows the exact same process. However, the other methods can be more efficient in many cases.
The quadratic formula might seem quite complex when you first come across it, but it can be broken down into smaller parts.
- The ± allows for the possibility of two solutions.
- The $b^2-4ac$b2−4ac under the square root sign is important as it will tell us how many solutions there are. This is known as the discriminant.
Practice questions
Question 1
Solve the equation $x^2-5x+6=0$x2−5x+6=0 by using the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$x=−b±√b2−4ac2a.
Write each solution on the same line, separated by a comma.
Question 2
Solve the following equation: $-6-13x+5x^2=0$−6−13x+5x2=0.
Write all solutions on the same line, separated by commas.