ACT Quadratics: Factoring, Roots, Vertex & Practice

Quadratics appear within ACT Algebra and Functions.

You should be comfortable moving between:

• standard form;
• factored form;
• vertex form;
• graph information.

For the full picture of every ACT Math topic, see the Enhanced ACT guide, or browse the rest of our ACT guides.

Standard form

[ ax^2+bx+c ]

Example:

[ x^2-7x+12 ]

Factored form

[ a(x-r_1)(x-r_2) ]

For:

[ x^2-7x+12 ]

factor:

[ (x-3)(x-4) ]

So the roots are:

[ x=3,\quad x=4 ]

Vertex form

[ a(x-h)^2+k ]

The vertex is:

[ (h,k) ]

Example:

[ y=2(x-5)^2-3 ]

Vertex:

[ (5,-3) ]

Because (a>0), the parabola opens upward and the vertex is a minimum.

Zero-product property

If:

[ (x-2)(x+7)=0 ]

then:

[ x=2\quad\text{or}\quad x=-7 ]

Quadratic formula

[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} ]

Useful when factoring is awkward.

Discriminant

[ b^2-4ac ]

This tells you the number of real solutions.

Positive

2 distinct real solutions.

Zero

1 repeated real solution.

Negative

No real solutions.

Example

[ x^2+4x+4=0 ]

Discriminant:

[ 4^2-4(1)(4)=16-16=0 ]

So there is one repeated real root.

Maximum and minimum problems

If:

[ P(t)=-3(t-4)^2+80 ]

the maximum value is:

[ 80 ]

at:

[ t=4 ]

You do not need to expand.

The same optimization idea — finding a maximum or minimum — turns up again in ACT statistics and probability questions about expected outcomes.

Intercepts

For:

[ y=(x-1)(x-6) ]

x-intercepts:

[ 1,\ 6 ]

For y-intercept, set x=0:

[ y=(-1)(-6)=6 ]

Original practice

1

Solve:

[ x^2-9x+20=0 ]

2

What is the vertex of:

[ y=(x+3)^2-5 ]

3

How many real solutions does:

[ x^2+2x+8=0 ]

have?

4

If:

[ y=-2(x-4)^2+10 ]

what is the maximum y-value?

5

A quadratic has roots 2 and -5. Which expression could represent it?

A. ((x-2)(x-5))
B. ((x+2)(x-5))
C. ((x-2)(x+5))
D. ((x+2)(x+5))

Answers

1. 4 and 5
2. (-3,-5)
3. 0 real solutions
4. 10
5. C

Common traps

• forgetting the sign change between factor and root;
• reading the vertex sign incorrectly;
• assuming every quadratic factors nicely;
• confusing x-intercepts with the y-intercept;
• forgetting that a negative leading coefficient means the parabola opens downward.

The same graphing instincts — reading vertices, intercepts and direction — come up again in ACT geometry and trigonometry.

Official source

• ACT Math Test Description: https://www.act.org/content/act/en/products-and-services/the-act/test-preparation/description-of-math-test.html

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