Standard and factored form of a quadratic

Warm-up

Three quick questions from earlier lessons.

Discover

$y = (x - 3)(x + 5)$ . Where does the graph cross the $x$ -axis?

At $x = 3$ and $x = -5$
At $x = -3$ and $x = 5$

Key idea

Two forms of the same quadratic.

Standard form

ax^2 + bx + c

shows the

y

-intercept

c

. Factored form

a(x - p)(x - q)

shows the

x

-intercepts

p

and

q

. Expand (multiply every term by every term) to go from factored to standard form.

Worked example

$(x + 4)(x - 2)$ .

x^2 - 2x + 4x - 8

x·x, x·(−2), 4·x, 4·(−2).

x^2 + 2x - 8

Collect.

x

-intercepts

-4

and

2

From the factors.

Your turn, guided

Your turn: expand $(x + 3)(x + 5)$ .

x^2 + 8x + 15

x² + 5x + 3x + 15.

Practice ★

Find the $x$ -intercepts of $y = (x - 6)(x + 2)$ .
Find the $x$ -intercepts of $y = (x - 7)(x + 1)$ .

Write $(x + 2)(x - 3)$ in standard form $ax^2 + bx + c$ .

Watch out!

Practice ★★

A quadratic with leading coefficient 1 crosses the $x$ -axis at $x = 9$ and $x = -4$ . Write it in factored form.
A quadratic with leading coefficient 1 crosses the $x$ -axis at $x = 7$ and $x = -6$ . Write it in factored form.

Exit ticket

Three questions to check you have got it.

Find the $x$ -intercepts of $y = (x - 6)(x + 2)$ .
Find the $x$ -intercepts of $y = (x - 7)(x + 1)$ .
Find the $x$ -intercepts of $y = (x - 3)(x + 5)$ .

Summary card

Standard and factored form

Factored form gives the $x$ -intercepts; standard form gives the $y$ -intercept.

Watch out: the factor $x + 5$ gives the intercept $-5$ .

End of the lesson

Next lesson: Graphs of quadratic functions: direction, intercepts, symmetry Back to the chapter

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