Quadratic Functions & Complex Numbers

Every quadratic function can be written in three equivalent forms. Each form reveals something different at a glance.

• Standard form: $y=ax^2+bx+c$  (reveals the $y$-intercept $c$)
• Vertex form: $y=a(x-h)^2+k$  (reveals the vertex $(h,k)$)
• Factored form: $y=a(x-r_1)(x-r_2)$  (reveals the $x$-intercepts $r_1,r_2$)

In all three, the number $a$ is the same: it controls direction and width.

For $y=ax^2+bx+c$:

• Direction: opens up if $a>0$ (has a minimum), down if $a<0$ (has a maximum).
• Axis of symmetry: the vertical line $x=-\dfrac{b}{2a}$.
• Vertex: $\left(-\dfrac{b}{2a},\ f\!\left(-\dfrac{b}{2a}\right)\right)$; this is the min or max point.
• $y$-intercept: $(0,c)$.  $x$-intercepts: solve $ax^2+bx+c=0$.

Factoring uses the Zero Product Property: if $A\cdot B=0$ then $A=0$ or $B=0$. Set the quadratic equal to $0$, factor, and solve each factor.

Square-root method works when there is no middle ($bx$) term, or when one side is a perfect square: isolate the square, then take $\pm$ the square root of both sides.

To complete the square on $x^2+bx$, add $\left(\dfrac{b}{2}\right)^2$. This makes

Use this to solve any quadratic or to convert to vertex form.

For $ax^2+bx+c=0$ with $a\neq0$,

The quantity under the radical, $D=b^2-4ac$, is the discriminant:

• $D>0$: two distinct real solutions.
• $D=0$: one repeated real solution.
• $D<0$: two complex (nonreal) solutions.

The imaginary unit is defined by $i=\sqrt{-1}$, so that $i^2=-1$. A complex number has standard form $a+bi$, where $a$ is the real part and $b$ is the imaginary part.

Powers of $i$ cycle with period $4$:

Add/subtract: combine real parts and imaginary parts separately.

Multiply: use FOIL, then replace $i^2$ with $-1$.

Conjugate: the conjugate of $a+bi$ is $a-bi$. Their product $(a+bi)(a-bi)=a^2+b^2$ is a real number.

Divide: multiply numerator and denominator by the conjugate of the denominator.

If the discriminant $D<0$, the graph has no $x$-intercepts and the two solutions are complex conjugates $p\pm qi$. Solve exactly as before—the negative under the radical just introduces $i$.

Quadratics model projectile height, areas, and any quantity with a single maximum or minimum.

• Projectiles: $h(t)=-16t^2+v_0t+h_0$ (feet, seconds). The peak is at the vertex time $t=-\dfrac{v_0}{2(-16)}$; the ground is where $h(t)=0$.
• Max/min: the extreme value always occurs at the vertex.

For $ax^2+bx+c=0$ with roots $r_1,r_2$, dividing by $a$ gives $x^2+\dfrac{b}{a}x+\dfrac{c}{a}=0$. Comparing with $(x-r_1)(x-r_2)=x^2-(r_1+r_2)x+r_1r_2$ shows

These Vieta's formulas let you read the sum and product of the roots straight off the coefficients—no solving required. Any symmetric function of the roots can then be rebuilt from these two numbers, e.g.

To construct a monic quadratic with prescribed roots $s_1,s_2$, use

When the new roots are transformations of old roots $r_1,r_2$, compute the new sum and product from Vieta's formulas of the original equation—you never need the roots themselves.

• Roots $r_1+k,\ r_2+k$: replace $x$ by $x-k$ in the original.
• Roots $kr_1,\ kr_2$: new sum $k(r_1+r_2)$, new product $k^2 r_1r_2$.
• Roots $\dfrac{1}{r_1},\dfrac{1}{r_2}$: reverse the coefficient list $a,b,c\to c,b,a$.

Treat the discriminant $D=b^2-4ac$ as a function of an unknown parameter to force a desired root behavior:

• Two distinct real roots: $D>0$.
• One repeated (double) root: $D=0$.
• No real roots (complex pair): $D<0$.

Setting $D=0$ also gives the tangency condition when a line meets a parabola in exactly one point (see below), and the boundary value where the number of real roots changes.

Plot $z=a+bi$ as the point $(a,b)$. Its modulus (distance from the origin) and argument (angle from the positive real axis) are

Key facts: $|z|^2=z\bar z$, moduli multiply ($|z_1 z_2|=|z_1||z_2|$), and multiplying by $i$ rotates a point $90^\circ$ counterclockwise. This geometric view explains why the powers of $i$ cycle: $i$ has modulus $1$ and argument $90^\circ$, so $i^n$ walks around the unit circle in $90^\circ$ steps.