Unit 6: Two Answers

Everything meets everything

To multiply out (x + 2)(x + 3), every term in the first bracket meets every term in the second.

That gives x^2 + 3x + 2x + 6, which tidies to x^2 + 5x + 6.

Notice where the numbers went: 2 and 3 added to make the middle, and multiplied to make the end. That pattern is about to be very useful.

Factoring runs it backwards

Factoring takes x^2 + 7x + 12 and asks: which two brackets made this?

You need two numbers that multiply to 12 and add to 7. Try the pairs for 12: 1 and 12, 2 and 6, 3 and 4. The last pair adds to 7.

So x^2 + 7x + 12 = (x + 3)(x + 4).

The one fact that solves them

If two numbers multiply to give zero, at least one of them *is* zero. Nothing else can do it.

So if (x - 2)(x - 3) = 0, then either x - 2 = 0 or x - 3 = 0.

That gives x = 2 or x = 3. Two answers, and both are genuinely right.

So: get zero on one side, then factor

That gives the whole method for a quadratic.

1. Move everything to one side so the other side is 0.
2. Factor what is left.
3. Set each bracket to zero.

x^2 + 5x + 6 = 0 factors to (x + 2)(x + 3) = 0, so x = -2 or x = -3.

When the x² has a coefficient

2x^2 + 7x + 3 will not factor into (x + a)(x + b) — the front needs a 2 somewhere.

It is (2x + 1)(x + 3). Check the middle: 6x + 1x = 7x. Correct.

The hunt is longer because the two numbers now multiply into different brackets, so try the possibilities in an orderly way rather than at random.

A tidier way to search

For 2x^2 + 7x + 3, multiply the front and back: 2 × 3 = 6.

Now find two numbers multiplying to 6 and adding to 7: that is 6 and 1.

Split the middle with them — 2x^2 + 6x + 1x + 3 — and factor in pairs: 2x(x + 3) + 1(x + 3), which is (2x + 1)(x + 3). No guessing at all.

Zero on one side, first, always

x^2 = 5x + 6 cannot be attacked yet. The zero-product rule needs a zero.

Move everything across: x^2 - 5x - 6 = 0.

Now it factors to (x - 6)(x + 1) = 0, so x = 6 or x = -1. Skipping the tidying step is the most common way to lose an answer.

A bracket equal to zero is not the same as a bracket equal to fifteen

x(x + 2) = 0 gives x = 0 or x = -2 at once, because a product is zero only when a part is.

But x(x + 2) = 15 tells you nothing directly. Plenty of pairs multiply to 15.

That is why the zero matters so much. Zero is the only number with that property.

Every quadratic draws the same curve

Plot y = x^2 and you get a parabola — a smooth U.

Wherever the curve crosses the bottom line, y is zero. So the crossing points are the solutions of the equation. A quadratic with two answers crosses twice, one answer touches once, and no real answers never reaches it at all.

The discriminant was telling you the shape all along.

Which way up, and where the turn is

If the x^2 term is positive the U opens upwards, and the curve has a lowest point. Negative, and it is an upside-down U with a highest point.

That turning point sits exactly halfway between the two roots — the parabola is symmetric.

So for roots at 2 and 8, the turn is at x = 5, no calculation needed beyond an average.

Two answers, one of them silly

A quadratic gives two answers. In a real story, often only one makes sense.

A rectangle cannot be -8 cm wide. A time cannot be negative. A number of people cannot be a fraction.

So solve properly, get both, then throw one away on purpose — and be able to say why.

When both answers are real

Sometimes both answers genuinely count.

A ball thrown up is at 20 m twice — once going up, once coming down. Both times are perfectly true.

So do not throw an answer away out of habit. Ask what it *means* first.

Completing the square

Most quadratics do not factor neatly. But every one of them can be rebuilt as a square.

x^2 + 6x + 5 is nearly (x + 3)^2, which is x^2 + 6x + 9. It is 4 short.

So x^2 + 6x + 5 = (x + 3)^2 - 4. Halve the middle number, square it, and patch the difference. Now the equation has only one x in it, and a square root finishes the job.

The formula, and where it comes from

Complete the square on the general quadratic ax^2 + bx + c = 0 and you get the same answer every time. That answer is the quadratic formula:

x = (-b + sqrt(b^2 - 4ac)) / (2a), and the same with a minus.

It is not magic. It is completing the square, done once, in advance, for everybody.

The bit under the root decides everything

b^2 - 4ac is called the discriminant, and you can read the answer off it before solving.

Positive — two different real answers.
Zero — the two answers land on the same spot, so just one.
Negative — no real answers at all… but complex ones, which is why unit 2 came first.