Factored Form

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

The form

A quadratic with real zeros can be written

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

Reading zeros and intercepts

From

the zeros are $-2$ and $5$, not $2$ and $-5$. Set each factor equal to zero rather than reading signs directly.

The intercept points are $(-2,0)$ and $(5,0)$.

Axis of symmetry

The axis lies halfway between the zeros:

This follows from parabola symmetry. Substitute the axis input into the equation to find the vertex output.

Worked example: full graph features

These landmarks support a reliable sketch.

Role of a

The coefficient $a$ controls opening and vertical scale. Positive $a$ opens upward; negative $a$ opens downward. Larger $|a|$ creates a narrower vertical stretch, while $0<|a|<1$ creates a wider vertical compression.

The zeros do not change when only $a$ changes.

Writing an equation from zeros and a point

If zeros $r,s$ are known, write

Substitute another point to find $a$. If zeros are $2$ and $7$ and the graph contains $(0,14)$:

so $a=1$.

Repeated zero

The form

has a double zero. The graph touches the $x$-axis at its vertex $(r,0)$ and turns around rather than crossing.

This corresponds to discriminant zero.

Converting to standard form

Expand using distribution:

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

Solving quadratic equations

If

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.

Modelling

Factored form is useful when a model's zero times are known, such as when a projectile begins and returns to ground level or when profit reaches break-even.

Restrict the quadratic to the meaningful interval and interpret which root occurs first.

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?

Related topics

• Quadratic Function
• Vertex Form
• Factoring Trinomials
• Discriminant