Tough Quadratic Simultaneous Equations for Paper 1 Non Calculator | Higher GCSE Maths 14th May 2026

Added: 2026-05-15

[00:00:00]Okay, so looking at a quadratic simultaneous equation. Again, doing this without calculator, it says we're going to solve algebraically these simultaneous equations.

[00:00:11]Now we have x^2 + y^2 = 25 and y - 3x is equal to 13.

[00:00:21]Now when we are solving quadratic simultaneous equations we do want to have a look first and see is there a way to make them both say y equals or x equals. But in this question here both x and y are both squared. So we're going to have to use the substitution method which means we need this second equation to say y equals or x equals. Now, we can go for either, but if we make it say x equals, at some point we're going to have to divide by 3. That means creating a fraction. So, it makes sense here to add 3x to both sides. So, that it says y equals. And if we do that just here, we get y = 13 add 3x. If you prefer, you can swap those pieces around and have it as y = 3x add 13. as long as the symbols with each piece are kept as they are.

[00:01:18]So from here we need to now find the value of y^ 2. If we know the value of y^ 2 we can substitute the expression for y^ 2 in place of the y^ 2. So y^2 means doing a double bracket. We're going to square the value of y. Squaring of course multiplying by itself. So 3x + 13 and another 3x + 13.

[00:01:46]We need to take our time with this. Make sure we get this correct. 3x * 3x gives you 9 x^ 2. We then have to do 3x * 13 and 13 * 3x. Each of those giving us 39x. And we have two of them.

[00:02:04]So 39x and another 39x in total gives us 78x and then 13 * 13 which is 169.

[00:02:18]So the expression that we get is 9 x^2 + 78x + 169 and we're going to put that in place of this y^ 2. So if we write this out, we now have x^2 + the 9 x^2 + 78x + 169 and that's now equal to 25.

[00:02:47]We have formed a quadratic equation. In order to solve a quadratic equation to find the value of x, we want this to equal zero. So there's two things that we can tidy up. We can join together the 1x^2 and the 9x^2 and we can minus 25 from both sides so that this equation is equal to zero.

[00:03:11]When we do that we get 10 x^2 + 78x and we'll get plus 169 takeway 25 is 144 and that's equal to zero.

[00:03:26]Now at this stage we need to factoriize.

[00:03:30]However, if all of the numbers that we can see in the quadratic equation have a common factor, meaning they all divide by the same number, then we can simplify the equation. Because it's equal to zero, we can divide both sides of the equation by a number, but zero won't change, so it's unaffected.

[00:03:54]So we can divide everything here by 2 definitely and that becomes 5x^2. Of course 78x is just half of the original 39x that we doubled in the first place.

[00:04:06]So 39x and 144 / 2 is 72 and that's equal to zero.

[00:04:17]Now there's no common factor that goes into 5 39 and 72. Five, of course, being a prime number, would have to go into 39 and 72, and it doesn't.

[00:04:28]So, we know when we factoriize this, we're going to have a 5x and a 1x.

[00:04:33][snorts] Now, there are different methods for factorizing quadratics. I use an inspection method. If you use your own method, absolutely fine. Stick with whatever method you prefer.

[00:04:44]72 has quite a lot of factors. We've got 1 and 72, 2 and 36.

[00:04:54]Beyond here, we'd have to be pretty good at knowing the factor pairs. So, we definitely need to or want to test any number along the way. Three does go in 24 times. Four goes in 18 times.

[00:05:10]Five doesn't go in. We've already established that. Six goes in into 72 12 times.

[00:05:20]Seven doesn't because we hit 70 and then 77. Eight does cuz it's eight less than 80. So 8 and 9. And now the numbers are right next to each other. So as long as I'm really confident I've found all the factors along the way there, I know that I have got every single factor pair of 72.

[00:05:39]I now need to find which factor pair is going to make 39 in the middle bearing in mind whatever number here is going to get multiplied by five. So we want to look down the list multiplying one of the numbers by five in each pair and seeing if I can make 39. Now I can't with either of the first two cuz I'd make five and 72 or a number that's far too big. 10 and 36. That's not going to work either. But the next one we get 15 and 24. I just need to make sure it works with the symbols. And this is all plus symbols. So it's going to work quite nicely. So that's going to work.

[00:06:19]So + three and plus 24.

[00:06:24]And there we go. We have it factorized.

[00:06:26]That gives us two solutions. We need to flip the symbol in the brackets.

[00:06:30]Hopefully we know solving equations. So x is equal to -3 and x is equal to -4 but that is going to be divided by 5.

[00:06:41]And there's our two x values.

[00:06:44]So we'll highlight those. We've got our first two solutions.

[00:06:49]When solving quadratic simultaneous equations, we also need to find that value of y. So we should have written down somewhere an equation that says y equals. And we did it right in the first step. So y is equal to 3x + 13.

[00:07:07]All we need to do is find the y value when x is equal to -3 and when x is equal to 24 over 5.

[00:07:19]I think we can see which of those is going to be more complicated.

[00:07:23]So we'll start with the one on the left.

[00:07:24]We have 3 * -3 and then add 13.

[00:07:31]3 * -3 is -9.

[00:07:34]So -9 add 13. And therefore we get y is equal to 4. And there is our pairing with the x is equal to -3.

[00:07:48]We just have one to go. And without a calculator it doesn't look like it's going to be the nicest. We have to do 3 * -4 over5 and then add 13 to it.

[00:08:02]So 3 * -4 over 5. Now 3 of course just being 3 over 1 we can multiply [clears throat] by 24. It's going to be -4 over 5. But 3 * 24 3 * 20 is 60. 3 * 4 is 12. So that's -72 over 5 + 13.

[00:08:29]Now I've got two options here. I can either write 13 as a fraction over 5. So 5 * 13 is 65. And I could write this as 65 over 5, which is 13.

[00:08:49]Or I could change -72 over 5 into a mixed number. 5 goes into 70 14 times.

[00:08:59]So it's 14 and two fths.

[00:09:02]It doesn't really matter which method you use here, but I need to use one of them. So, I'm going to go with the - 72 over 5 and add to it 65 because then I only have to find the difference between 72 and 65, which is 7. So, it's - 72 add 65 on the numerator there, which would give us -7 and that's in terms of fifths. So, it's - 7 fths and that would be my final solution there. But again, different methods that you could use to get that -75ths. Again, quite like the idea of turning 72 over 5 into a mixed number and then just adding 13.

[00:09:46]But there we go. That is a method that you could use. If you prefer a different method, of course, you can use that. But that is how we would go about solving quite a complex quadratic simultaneous equation and all of those steps there without a calculator. If you want all nine questions like this in one place, then grab my revise 9 revision grids completely free. Click the link in the description, create your free account, and unlock them inside upgrade or TGMT4.

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