1. The quadratic formula

2. Solving quadratic equations
What does it mean, both graphically and algebraically, to solve an equation such as 2x2 + x – 1 = 0 ?
Graphically, solving the equation means finding the intersection of the curve y = 2x2 + x – 1 and the line y = 0 (the x-axis).
Algebraically, solving the equation means finding the values of x that make it true.
We can do this by factoring the equation, completing the square, or trial and error. y
(–1, 0)
(0.5, 0)
We can also use the quadratic formula. x
y = 2x2 + x – 1

3. The quadratic formula
Any quadratic equation of the form ax2 + bx + c = 0 can be solved by substituting the values of a, b and c into the formula:
–b ± b2 – 4ac
x = 2a
This equation can be derived by completing the square on the general form of the quadratic equation.
Teacher notes
The quadratic formula is a complex formula, so stress that some practice is needed to use it correctly. Ask students if they can see how this formula will lead to two solutions. More able students could be asked to complete the square for the general quadratic equation to derive the formula. This derivation is given in the flash activity on the next slide.

4. The determinant
The expression under the square root sign in the quadratic formula, b2 – 4ac, shows how many solutions the equation has.
This expression is sometimes called the determinant.
x =
–b ± b2 – 4ac 2a
When b2 – 4ac is positive, there are two roots.
Teacher notes
The difference between a solution and a root can be difficult for students to understand. In order to emphasize the link with the graph, we have here used roots. Another way of describing this, which is helpful as students go on to math at higher levels, is to say that when b2 – 4ac is positive, there are two unique real solutions, when b2 – 4ac is equal to zero, there are two identical solutions, and when b2 – 4ac is negative, there are two non-real solutions.
Mathematical practices
7) Look for and make use of structure.
Students should see that the value of the expression b2 – 4ac determines how many solutions the equation has. They should realize that this means that expression can be used immediately on being given a quadratic function to determine how many roots it has.
When b2 – 4ac is equal to zero, there is one root.
When b2 – 4ac is negative, there are no roots.

5. The roots of a graph
We can demonstrate each of these possibilities using graphs.
Remember, if we plot the graph of y = ax2 + bx + c, the solutions to the equation ax2 + bx + c = 0 are given by the points where the graph crosses the x-axis. y x
b2 – 4ac is positive y x
b2 – 4ac is zero y x
b2 – 4ac is negative
Mathematical practices
7) Look for and make use of structure.
Students should see that the value of the expression b2 – 4ac determines how many solutions the equation has. They should realize that this means that expression can be used immediately on being given a quadratic function to determine how many roots it has.
two solutions
one solution
no real solution