Lesson 05 · General solver
Quadratic Formula & Discriminant
Use one general solving method for any quadratic in standard form, and read the discriminant first to predict how many real roots the graph can have.
The learner question
How can one formula solve every quadratic equation?
The quadratic formula packages the completing-the-square process for ax² + bx + c = 0. Its square-root input also reveals the root geometry before we finish calculating.
The formula
x = (−b ± √(b² − 4ac)) / 2a
Write the equation as ax² + bx + c = 0 before identifying a, b, and c.
Worked model
Solve 2x² + 3x − 2 = 0 without guessing factors.
01Identify coefficients
a = 2, b = 3, c = −2
Signs belong to their coefficients.
02Compute the discriminant
Δ = 3² − 4(2)(−2) = 25
Positive Δ predicts two real roots.
03Substitute as one fraction
x = (−3 ± 5) / 4
The whole numerator is divided by 2a = 4.
04Evaluate both branches
x = 1/2 or x = −2
The plus and minus create the two roots.
Δ > 0
two distinct real roots
The square root has positive magnitude, so plus and minus separate.
Δ = 0
one distinct real root
Both formula branches collapse to the same value.
Δ < 0
no real roots
The square root leaves the real-number system; the parabola misses the x-axis.
Discriminant forecast
Predict the root geometry before solving.
vertex (2.5, -0.25)
axis x = 2.5
Coefficient map
x² − 5x + 6 = 0
a
1
b
-5
c
6
Discriminant
Δ = b² − 4ac = 1
Sign discipline: if b is negative, then −b is positive. Keep parentheses around coefficient substitutions until the arithmetic is finished.
Check transferFresh cases, same underlying idea
Quadratic Formula & Discriminant check1 / 3