Quadratic function
Roots and delta
Roots are x values for which the function equals 0. In a quadratic function, their number depends on delta.
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How do you calculate delta?
For the function f(x) = ax² + bx + c we first calculate the discriminant of the quadratic trinomial:
Delta formula
Δ = b² - 4ac
The sign of delta tells you whether the parabola crosses the X-axis, touches it, or lies entirely above or below it.
X-axis intercept
On the graph, the roots are the points where the parabola crosses the X-axis. For example, x² - 2x - 3 has roots x = -1 and x = 3.
Interpreting delta
Delta is a quick test for the number of solutions of a quadratic equation.
Δ > 0
Two roots
The parabola crosses the X-axis twice.
Δ = 0
One root
The parabola touches the X-axis at the vertex.
Δ < 0
No roots
The parabola does not cross the X-axis.
Ready formulas
Roots of a quadratic function
We use these formulas when delta is non-negative.
When Δ = 0, both formulas give the same result: x₁ = x₂ =-b2a.
Example: f(x) = x² - 2x - 3
Step 1
a = 1, b = -2, c = -3
We list the coefficients from the general form.
Step 2
Δ = (-2)² - 4 · 1 · (-3) = 16
Delta is positive, so there will be two roots.
Step 3
x₁ = (2 - 4) / 2 = -1
We substitute into the formula with minus before the square root.
Step 4
x₂ = (2 + 4) / 2 = 3
We substitute into the formula with plus before the square root.
The roots are x = -1 and x = 3, so the factored form is f(x) = (x + 1)(x - 3).
What next?
After mastering delta, it is worth practicing quadratic equations and inequalities, because roots determine the boundaries of intervals there.