For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a. It solves every quadratic.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Standard form first
Move everything to one side before reading a, b and c. Reading them from an unarranged equation is the usual error.
The discriminant
The expression under the root, b² − 4ac, predicts the roots before you compute them.
What it predicts
Positive gives two real roots, zero gives one repeated root, and negative gives no real roots.
A useful extra
A perfect square discriminant means the quadratic would have factored. That is worth knowing before starting.
Which method
Try factoring first, since it is faster when it works. Use the formula whenever factoring stalls.
The formula is completing the square, done once
Applying the technique to the general ax² + bx + c = 0 produces x = (−b ± √(b² − 4ac))/(2a). The formula exists so nobody repeats that derivation, but doing it once makes it unforgettable.
The discriminant predicts the answer
b² − 4ac positive gives two real roots, zero gives one repeated root, negative gives two complex roots. Computing it first tells you what kind of answer to expect.
The two standard errors
Dividing only part of the numerator by 2a, and losing the minus on −b. Writing the formula out in full before substituting, rather than substituting from memory, prevents both.
Choosing a method
Factor when it factors easily, use the formula when it does not, complete the square when you need the vertex. Having three methods and choosing sensibly is the actual skill.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Ines Use the Formula
Ines solves x² − 4x + 1 = 0, which does not factor.
- Step 1
She reads a = 1, b = −4 and c = 1 from standard form.
Step 4: Your Turn
Practice makes it stick.
The Discriminant
Problem 1 of 2
x² − 4x + 1 = 0. What is the discriminant?
The Prediction
Problem 2 of 2
x² + 2x + 5 = 0. What is the discriminant?
Predict and Solve
1 of 8
x² − 6x + 5 = 0. Discriminant?
2 of 8
Same equation. Larger root?
3 of 8
x² − 4x + 4 = 0. Discriminant?
4 of 8
How many distinct real roots does that have?
5 of 8
x² + x + 3 = 0. Discriminant?
6 of 8
How many real roots does that have?
7 of 8
Sort each discriminant by what it predicts.
Tap something to move it.
- Empty
- Empty
8 of 8
2x² − 8x + 6 = 0. Larger root?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x² + 3x + 2 = 0. What is the discriminant?
Question 2 of 2
A negative discriminant means what about the graph?
What You Learned
- The quadratic formula solves every quadratic.
- The discriminant predicts two real roots, one, or none.
- Try factoring first; use the formula when it stalls.