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a + b = 6,

a - b = 2.

A way of solving these is to add the two equations together, so

a + a + b - b = 6 + 2

or, more simply, 2a = 8 (because the +b and -b cancel each other out).

From there you can calculate that a = 4 and, because a + b = 6, b must equal 2. Which is verified by the second equation, 4 - 2 = 2.

The principle remains the same regardless of how many unknowns you have:

a + b + c = 24,

a + b - c = 16,

2a + b = 32.

Add the first two equations together and you get 2a + 2b = 40 (because this time the c’s cancel each other out).

Now look at the third equation. It’s very similar to the sum of the first two. Subtract one from the other:

(2a + 2b) - (2a - b) = 40 - 32.

The a’s cancel each other out, so 2b - b (in other words, b) = 8.

Go back to the third equation, which contains only a’s and b’s, and substitute 8 for b:

2a + 8 = 32.

Deduct 8 from each side of the equation to give

2a = 32 - 8 = 24,

which means that a = 12.

Now go back to the first equation and substitute both a and b:

12 + 8 + c = 24,

20 + c = 24,

c = 24 - 20 = 4.

Verify this by going to the second equation:

12[a] + 8[b] - 4[c] = 16,

which is true.

☞ QUADRATIC EQUATIONS

These are more complex again, because they involve a square—that is, a number multiplied by itself and written with a raised 2 after it—so 16 is 42, and 36 is 62. Thus, 4 is the square root of 16, and 6 is the square root of 36. The symbol for a square root is √. Actually, (-4)2 is also 16, so 16 has two square roots: +4 and -4. Any positive number has two square roots. A negative number doesn’t have any square roots at all, because if you multiply a negative by a negative, you get a positive.

An algebraic expression can also be a square: the square of a + 4 is (a + 4) x (a + 4). You do this by multiplying each of the elements in the first bracket by each of the elements in the second:

(axa) + (ax4) + (4xa) + (4x4)

= a2 + 8a + 16.

To solve a quadratic equation, you need to turn both sides of it into a perfect square, which is easier to explain if we look at an example:

a2 + 8a = 48.

The rule for “completing the square” in order to solve a quadratic equation is, “Take the number before the a, square it, and divide by 4.” For example, 8 squared (64) divided by 4 is 16, so we add that to both sides; reassuringly, we already know that adding 16 to this equation will create a perfect square, because we just did it in the previous equation:

a2 + 8a + 16 = 48 + 16 = 64 .

Taking the square root of each side gives:

a + 4 = 8 (because 8 is the square root of 64).

Again, we know that a + 4 is the square root of a2 + 8a + 16, because it was part of the sum we did on the previous page. Anyway, we now have a simple sum to establish that a = 4.

Wait a minute, though. Taking the square root of both sides of an equation is not allowed. Why is this? Because a positive number like 64 has two square roots, +8 and -8. So the truth of the matter is that actually

a + 4 = +8 or -8,

so a equals either +4 or -12.

Although this example is an easy one, the beauty of algebra is that the same principle applies whatever the numbers involved. So, to repeat: The rule for “completing the square” in order to solve a quadratic equation is, Take the number before the a, square it, and divide by 4.)

So if your equation is

a2 + 12a + 14 = 33,

you first simplify the equation by getting rid of the 14. Subtract it from both sides to leave:

a2 + 12a = 33 - 14 = 19.

Square the 12 to give 144, divide by 4 to give 36, and—as always—add that to both sides:

a2 + 12a + 36 = 19 + 36 = 55.

The square root of that gives you

a + 6 = 55 = (approximately) 7.4, or, of course, -7.4.

Deduct 6 from each side to leave the simple statement a = 1.4 or -13.4.

You can check that this is right by going back to the original equation and putting in a = 1.4:

a2 + 12a + 14 = 33 becomes

(1.4 x 1.4) + (12 x 1.4) + 14 = 1.96 + 16.8 + 14

(near enough for the purposes of this exercise)

= 2 + 17 + 14 = 33.

QED, as they