Questions on Quadratics Trivia Quiz

Answer: Factorising

Factorising is a way of splitting up the original equation into, as the name suggests, factors or constituents. In the example given, x^2 + 9x + 20 (^ meaning to the power of), the factors would be (x+4) and (x+5).
If you use the method of FOIL (First, Inner, Outer, Last) to multiply these two factors you will see that you will go back to the original equation.

(x+4)(x+5) - x multiplied by x is (x^2), 4 multiplied by x is (4x), 5 multiplied by x is (5x) and 4 multiplied by 5 is (20). As you can see, the answers in the brackets are the components of my original quadratic equation.

Answer: (x+3) and (x-2)

The answer is (x+3) (x-2) and here is the proof. Using the method of FOIL - x multiplied by x is (x^2), 3 multiplied by x is (3x), -2 multiplied by x is (-2x) and 3 multiplied by -2 is (-6). So, you are left with the equation x^2 + 3x - 2x - 6. This can be further simplified by using basic algebra as 3x - 2x equals 1x (the co-efficient of x which is one, is pretty redundant in this case and is therefore simply expressed as just x).

Answer: (x-3) and (x-6)

The common mistake made in this sort of equation where the co-efficient of x is negative whilst the number on its own is positive is the multiplication of two negative numbers. One of the fundamental rules of arithmetic is that a negative multiplied by a negative equates to a positive.

Answer: (3x+3) and (x+2)

These are slightly more difficult to solve. There are various methods available such as cross multiplication and so on. However, the method I am going to use to explain this is quicker as you don't have to go by trial and error to attain your roots (answers).

The original equation was 3x^2 + 9x + 6. Quadratic questions follow a general rule which is ax^2 + bx + c. This indicates that x^2 is component a, the x term is component b and the stand alone number is component c.
The first step would be to multiply the co-efficient of component a and component c together which would equal, in this instance, 18 (3 x 6). The next step would be to find two numbers that multiply together to make that total of 18 and also add together to the sum of the co-efficient of the component b which, in this instance is equal to 9. The only two numbers able to do so are 6 and 3. So, you now split component b (9x) into the two new co-efficient which are 6 and 3 (6x + 3x). Your new formula is 3x^2 + 6x + 3x + 6. The next step would be to factorise this equation in pairs, 3x^2 + 6x and 3x + 6. If we take 3x^2 + 6x, we need to find the highest common factor which, in this case, is 3x. If you divide 3x^2 and 6x by 3x you get 3x(x+2). If we then take the second half which is 3x + 6, you will find that the highest common factor is 3. Divide both 3x and 6 by 3 and you get 3(x+2). If we put both of these factors together we get 3x(x+2) + 3(x+2). You will notice that both terms inside the brackets/parentheses are x+2 - this is a clear indication that you are on course for the correct answer. You now group the terms outside the brackets and the terms inside the brackets together. You will get (3x+3) and (x+2), et voila, it matches the answer given.

This seems like a long winded and drawn out process, but, with a bit of practice it is possible to solve the problem within 30 seconds.

Answer: x = 2, x = 2

The first stage of this equation is factorisation. You are looking for two numbers that multiply to give 4 and add to give -4. After a bit of mental arithmetic you should come up with -2 and -2 as -2 + -2 = -4 and -2 x -2 = 4. As this is an equation the following step is to solve the equation to attain the roots: x - 2 = 0 and x - 2 = 0. Using basic algebra, you add two to each side of the equations and you come up with the answers x = 2 and x = 2. Yes, these answers are the same -- x = 2 is a double root.

Answer: x = -4, x = 2

The first step is factorisation. What two numbers add together to make 2 and multiply together to produce -8. After that dreaded mental arithmetic you arrive at the terms (x+4) and (x-2). As the equation states the components add up to a total of zero, you reach the following: x + 4 = 0 and x - 2 = 0. Basic algebra attains the answers x = -4 and x = 2.

Answer: x^2 + 5x - 24

FOIL (First, Inner, Outer, Last) is a very good mnemonic for remembering how to expand brackets/parentheses. (x+8)(x-3): x multiplied by x = (x^2), 8 multiplied by x = (8x), -3 multiplied by x = (-3x) and 8 multiplied by -3 = (-24).
Put these together and you get: x^2 + 8x - 3x - 24. This can be simplified to make: x^2 + 5x - 24.

Answer: x^2 - 10x + 24

This one can be a little tricky especially when you are doing the 'last' stage of FOIL. First: x multiplied by x = (x^2). Outer: -4 multiplied by x = (-4x). Inner: -6 multiplied by x = (-6x). Last: -6 multiplied by -4 = (24). NOT -24!
Put all of these together and you get x^2 - 4x - 6x + 24. Simplify and you get x^2 - 10x + 24.

Answer: b^2 - 4ac

The full equation is x = (-b ± [square root b^2 - 4ac]) / 2a. Quite a mouthful, however, if known by heart makes the topic of quadratics a lot easier! Bearing in mind that quadratic expressions follow the pattern ax^2 + bx + c = 0, the quadratic formula is just a device to substitute the letters (a, b and c) with numbers given in the expression. If the equation is, for instance, (1)x^2 + 5x + 6: a = 1, b = 5 and c = 6. Simply substitute the letters for those numbers and the rest is basic arithmetic.

Answer: u

If the quadratic expression is positive (indicated by the co-efficient of x^2 being positive) then a 'u' shaped curve will be given. This is known as a parabola. If the quadratic expression is negative (indicated by the co-efficient of x^2 being negative) then an 'n' shaped curve will be given.

Thanks for playing this quiz - if your head is hurting, you are not alone! Hope you enjoyed.