ALGEBRA

Quadratic Equations

Quadratic Equations

General form

A quadratic equation is one which contains an x2 term. The general form of a quadratic is

ax2 + bx + c = 0

b and c can take any value, including zero. a can be any number except for zero, becuase if a were zero we wouldnt have an x2 term and so we wouldnt have a quadratic equation.

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Factorising

Remember, factorising is a way to simplify an expression. It involves put an equation into brackets, the opposite of expanding brackets.

The easiest way to do this is to look for the common factor in the equation.

For example, take 12x + 18.

The common factor between 12 and 18 is 6. So you put 6 outside the brackets and divide both numbers in the equation by 6:

6(2x + 3)

Quadratic equations can also be factorised.

For example: factorise 6g2 – 2g.

The common factor is 2 and g. This means that you divide both parts by 2g then put 2g at the front of the brackets:

2g(3g – 1)

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Factorising quadratic equations (Higher Tier)

mathsWhen you factorise a quadratic equation you need to put it back into the brackets. In the case where you have all 3 terms, then you will need to put in into two sets of brackets.

For example: factorise x2 – 10x + 24.

First of all you need to figure out two numbers that add to 10 and multiply to 24:

Factor pairs of 24 are:

  • 1 and 24
  • 2 and 12
  • 3 and 8
  • 4 and 6

4 and 6 add up to 10 so this is the pair you need to put into the brackets:

(x +4) (x + 6)

You can check your answer by expanding the brackets again:

(x +4) (x + 6)

= x2 + 4x + 6x + 24

= x2 + 10x + 24

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The difference of two squares

mathsYou might be faced with a quadratic equation that has no x term, only x2 and a number.

However, you can factorise the expression as before, it just means that the two coefficients of x must add up to make 0.

For example, factorise x2 – 49.

To get the – 49, we need to find two numbers that multiply to give -49 and add to make 0. The only pair that will work are -7 and 7.

(x – 7) (x + 7)

Factorising an expression like this is called the difference of two squares. However, not all quadratic expressions without an x term can be factorised. To factorise, the expression must contain an x2 term minus a square number such as 4, 9, 16, 25 etc.

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Solving quadratic equations (Higher Tier)

There are three main ways you can solve a quadratic equation:

      • factorising
      • completing the square
      • formula

To solve a quadratic equation by factorising you first need to write it out in the form:

ax2 + bx + c = 0

Then you can put it into brackets and figure out the possible solutions for x.

For example: solve the equation x2 + 2x – 8 = 0

So, find the two numbers that add up to 1 and multiply to make -3:

(x + 4) (x – 2) = 0

This means that:

(x + 4) = 0 or (x – 2) = 0

So, x = -4 or x = 2

You can check your answer is correct by substituting your answers into the original equation:

x= -4:

x2 + 2x – 8 = 0

-42 + (2 x -4) – 8 = 16 – 8 – 8 = 0

x = 2

22 + (2 x 2) – 8 = 4 + 4 – 8 = 0

If a quadratic equation won’t factorise then you can solve it by completing the square.

A good way to go about this is to write then algebraic expression as a square plus another term.

For example, x2 – 8x + 16.

You can factorise this equation to make:

(x – 4)2 so x = 4

However, you may get an equation in which you get another term as well. The other term can be calculated by dividing the co-efficient of x by 2 and then squaring it.

For example, work out x2 + 6x

The co-efficient of x is 6. So, if you divide 6 by 2 and then square it you get 9.

Therefore, the answer is: (x + 3)2 – 9

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