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secondary 3 | A Maths
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not quite sure how to do this question! I keep getting the incorrect answer. help would be greatly appreciated asap thank you !
Date Posted: 2 years ago
Views: 153
Solving for x is it?
A cubic equation is looming...
Of course, equations are simpler to solve (at least to our eyes) when there are no fractions.
4x (x + 1) = 43 - 20/x
4x² + 4x = 43 - 20/x
We multiply both sides by x to eliminate that fraction.
4x³ + 4x² = 43x - 20
4x³ + 4x² - 43x + 20 = 0
We have achieved what we need. The cubic equation. Unlike quadratic equations, there is no immediate way to solve cubic equations in general (at least at your level, though I am not aware of one).
We need to actually guess a value of x out. Fortunately, this can be done on your calculator (depending on whether you use Casio fx 96, Casio fx 97, Sharp 531 or others).
Typically, the number we guess are factors of the constant term.
Let me make a guess. Maybe x = -4?
Let f(x) = 4x³ + 4x² - 43x + 20.
When x = -4, f(-4) = 4 (-4)³ + 4 (-4)² - 43 (-4) + 20 = -256 + 64 + 172 + 20 = 0.
Yes! I got it on my first try!
(Ok, I cheated, I found it from a graphing software online)
So, we guessed x = -4 correctly, so by the Remainder/Factor Theorem, x + 4 is a root of f(x).
Wait, I still have more to post.
Of course, equations are simpler to solve (at least to our eyes) when there are no fractions.
4x (x + 1) = 43 - 20/x
4x² + 4x = 43 - 20/x
We multiply both sides by x to eliminate that fraction.
4x³ + 4x² = 43x - 20
4x³ + 4x² - 43x + 20 = 0
We have achieved what we need. The cubic equation. Unlike quadratic equations, there is no immediate way to solve cubic equations in general (at least at your level, though I am not aware of one).
We need to actually guess a value of x out. Fortunately, this can be done on your calculator (depending on whether you use Casio fx 96, Casio fx 97, Sharp 531 or others).
Typically, the number we guess are factors of the constant term.
Let me make a guess. Maybe x = -4?
Let f(x) = 4x³ + 4x² - 43x + 20.
When x = -4, f(-4) = 4 (-4)³ + 4 (-4)² - 43 (-4) + 20 = -256 + 64 + 172 + 20 = 0.
Yes! I got it on my first try!
(Ok, I cheated, I found it from a graphing software online)
So, we guessed x = -4 correctly, so by the Remainder/Factor Theorem, x + 4 is a root of f(x).
Wait, I still have more to post.
So now that we have established this factor (x + 4), we then proceed to do long division or compare coefficients (or some other suitable, logical approach).
(For this particular question, the three roots are easily found as rational numbers, so you can guess all three out; in general, for cubic equations at your level, at least one root is rational; it can be one rational and two irrational, one rational and two unreal or two rational only)
Recall the long division which you did on the previous question? We do a likewise approach.
After long division, we get f(x) = (x + 1) (4x² -12x + 5) which can be further factorised into f(x) = (x + 1) (2x - 1) (2x - 5).
So essentially, solving the question is akin to solving the equation (x + 1) (2x - 1) (2x - 5) = 0.
Our solutions are x = -1, x = 1/2 and x = 5/2.
(For this particular question, the three roots are easily found as rational numbers, so you can guess all three out; in general, for cubic equations at your level, at least one root is rational; it can be one rational and two irrational, one rational and two unreal or two rational only)
Recall the long division which you did on the previous question? We do a likewise approach.
After long division, we get f(x) = (x + 1) (4x² -12x + 5) which can be further factorised into f(x) = (x + 1) (2x - 1) (2x - 5).
So essentially, solving the question is akin to solving the equation (x + 1) (2x - 1) (2x - 5) = 0.
Our solutions are x = -1, x = 1/2 and x = 5/2.
alright understood ! thank you very much!!
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How this helps
Date Posted:
2 years ago
U can use mode 3 4 in ur casio calculator(mine is fx-96SG) to find out factors easily
Depends on the calculator model.
Ye
Newer batches of students use the Casio FX 97 SG plus, and the commands are different (if I recall, on this calculator it's MODE --> 5 --> 2 --> input 3 for a degree 3 polynomial.
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