Quadratic with zeros -4 and 1 through (2, -12) Step by Step

Step by step

The question

Question: write a quadratic with zeros −4 and 1 through (2, −12).

zeros : −4, 1

through : (2, −12)

A zero of f is an x that makes f(x) equal 0.

zero: an x where f(x) = 0

1 Each zero gives a bracket

If c is a zero, then x minus c is a factor.

Put c in, and that bracket is 0, so all of f(x) is 0.

zero c → bracket (x − c)

The zero −4 gives x − (−4).

Minus a minus is plus: x + 4.

x = −4 → x − (−4) = x + 4

Next, 1 gives x − 1.

x = 1 → x − 1

Careful!

The sign flips.

The zero −4 gives x + 4, because that is 0 when x is −4.

Careful: not zero −4 → (x − 4) ✗

2 Write f(x)

The zeros fix the brackets, but f can be stretched: a times the brackets.

The point tells us a.

f(x) = a(x + 4)(x − 1)

3 Find a

Put x = 2 and f(x) = −12 in.

−12 = a(2 + 4)(2 − 1)

Work out each bracket: 6 and 1.

−12 = a(6)(1)

6 times 1 is 6.

So −12 = 6a.

−12 = 6a

To remove times 6, we divide both sides by 6.

−126 = 6a6

6 divided by 6 is 1.

So just a is left.

So a = −2.

a = −2

We put a in front of the brackets.

f(x) = −2(x + 4)(x − 1)

4 Multiply out

Each part of the first bracket times each part of the second.

each part × each part

(x + 4) times (x − 1).

Each part times each part.

= −2[x² − x + 4x − 4]

Like terms together: x² + 3x − 4.

= −2(x² + 3x − 4)

−2 times each part.

= −2x² − 6x + 8

So f(x) = −2x² − 6x + 8.

f(x) = −2x² − 6x + 8

5 Check

Let's check: put each zero in.

First x = −4.

We get 0.

f(−4) = −2(−4)² − 6(−4) + 8 = −32 + 24 + 8

= 0

And x = 1.

We get 0.

f(1) = −2(1)² − 6(1) + 8 = −2 − 6 + 8 = 0

And the point: put x = 2 in.

We get −12, as it should be.

f(2) = −2(2)² − 6(2) + 8 = −8 − 12 + 8

= −12

Let's look back.

First: each zero gives a bracket.

Then: write f(x).

Then: find a.

Then: multiply out.

Last: check.

Answer