Solving the Quadratic Equation and the Discriminant

Every quadratic equation $ax^2+bx+c=0$ ($a\neq 0$) can be solved by the same completing-the-square trick, and it always produces the two roots

The key idea in the derivation: multiply through by $4a$ and rearrange so the left side becomes a perfect square, $(2ax+b)^2 = b^2-4ac$; taking square roots and isolating $x$ then gives the formula above. Because a quadratic always has two roots (possibly equal, possibly complex), this single formula settles every quadratic equation there is — no case-by-case guessing is needed.

The expression under the square root, $b^2-4ac$, controls everything about the character of the roots and is important enough to have its own name and symbol: the discriminant,

When $a,b,c$ are real, $\Delta$ splits the roots into exactly three families:

• $\Delta = 0$: the two roots collapse into one repeated (double) root, $\alpha=\beta=-\dfrac{b}{2a}$.
• $\Delta > 0$: the roots are real and distinct.
• $\Delta < 0$: the roots are non-real complex numbers, and they always come as a conjugate pair — if $p+iq$ is a root, so is $p-iq$.

When $a,b,c$ are additionally rational, $\Delta$'s sign refines further into whether the roots are rational or irrational: $\Delta=0$ gives equal rational roots, $\Delta>0$ with $\Delta$ a perfect square of a rational number gives distinct rational roots, and $\Delta>0$ with $\Delta$ not a perfect square gives roots that are conjugate surds (like $2+\sqrt3$ and $2-\sqrt3$).

Worked example. Find the roots of $x^2 - 2x + 5 = 0$ and describe their nature. …