Factored Form

Factored form y=a(x−r)(x−s) reveals a quadratic's zeros, symmetry, opening, and scale.

Structure → factor → verify. An expression is scanned for common structure, rewritten as factors, and checked by multiplication. The return arrow makes verification part of the method rather than an optional final step.

Factored form makes the $x$-intercepts visible before a single expansion is performed.

The form

A quadratic with real zeros can be written

y=a(x-r)(x-s),\qquad a\ne0.

The zeros are $x=r$ and $x=s$ because either factor becomes zero. If $r=s$, the repeated factor creates one touching intercept.

Worked example: full graph features

These landmarks support a reliable sketch.

Converting to standard form

Expand using distribution:

$$ a(x-r)(x-s) =a[x^2-(r+s)x+rs]. $$

Standard form reveals the $y$-intercept directly and supports completing the square or quadratic formula methods.

Solving quadratic equations

$$ a(x-r)(x-s)=0, $$

the zero-product property gives $x=r$ or $x=s$. The equation must equal zero before individual factors can be set to zero.

Do not use zero-product reasoning when the product equals a nonzero number.

Common mistakes

Reading the factor sign directly. $(x+3)=0$ gives $x=-3$.

Finding the axis by adding but not dividing by two. It is the midpoint of roots.

Ignoring $a$ when finding the vertex output. Substitute into the complete formula.

Setting factors to zero when their product is not zero. Zero-product property has a condition.

Assuming every quadratic has real factored form. Negative discriminant gives no real linear factors.

Quick self-check

What values make each factor zero?
Is the axis the average of the roots?
What vertex output follows at that axis?
Does $a$ match the opening and width?
Does $x=0$ give the correct $y$-intercept?
Do expansion and graph features agree?