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MathMagic

Quadratic Equations

From the shape of a parabola to the formula that finds its roots: what a, b and c each control, and what the discriminant tells you.

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  • Functions and Graphs (planned)

step 01 / 07 · A parabola from three numbers

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A parabola opening upward with its lowest point at the origin.

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  1. A parabola from three numbers

    Every quadratic function has the same shape of rule:

    f(x)=ax2+bx+cf(x) = ax^2 + bx + c

    Three numbers, aa, bb and cc, decide everything. Start with the plainest case, a=1a=1, b=0b=0, c=0c=0, which is just f(x)=x2f(x)=x^2: a bowl resting on the origin.

  2. a sets how narrow it is

    Raise aa to 22 and every height doubles, so the bowl narrows. The dashed curve is the original x2x^2 for comparison.

    A larger ∣a∣|a| always means a steeper, narrower parabola.

  3. A negative a flips it

    Set a=−1a=-1 and the parabola turns over: now it has a highest point instead of a lowest one.

  4. c slides it up and down

    Back to a=1a=1. Now let c=3c=3. The whole curve rises by 33 and crosses the yy-axis at (0,3)(0,3).

    That is the one job of cc: it is the height where the curve meets the yy-axis, because f(0)=cf(0)=c.

  5. b moves the axis of symmetry

    With b=−2b=-2 the bowl slides right. Every parabola is symmetric about the vertical line through its vertex, at

    x=−b2ax = -\frac{b}{2a}

    Here that is x=1x = 1, and the vertex sits at (1,2)(1, 2).

  6. Roots: where it meets the x-axis

    The roots are the xx-values where f(x)=0f(x)=0. The quadratic formula finds them:

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

  7. When there are no real roots

    The quantity under the root, Δ=b2−4ac\Delta = b^2 - 4ac, decides how many real roots exist:

    • Δ>0\Delta > 0: two roots
    • Δ=0\Delta = 0: one root, where the vertex touches the axis
    • Δ<0\Delta < 0: none, so the curve never reaches the xx-axis

    For x2−2x+2x^2-2x+2 we get Δ=4−8=−4\Delta = 4-8 = -4. The vertex (1,1)(1,1) floats above the axis.