Given the equation: $$x^4 - 8x - 29 = (x^2 + a)^2 + b,$$ where $a$ and $b$ are constants. (a) Find the value of $a$ and the value of $b$. (b) Hence, or otherwise, show that the roots of $$x^4 - 8x - 29 = 0$$ are $c imes rac{1}{2} ext{ and } d imes rac{1}{2}$, where $c$ and $d$ are integers to be found.

A-Level Edexcel Maths Pure Circles: Given the equation: $$x^4

Given the equation: $$x^4 - 8x - 29 = (x^2 + a)^2 + b,$$ where $a$ and $b$ are constants - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 1

Question 5

Given the equation: $$x^4 - 8x - 29 = (x^2 + a)^2 + b,$$ where $a$ and $b$ are constants. (a) Find the value of $a$ and the value of $b$. (b) Hence, or ot... show full transcript

Worked Solution & Example Answer:Given the equation: $$x^4 - 8x - 29 = (x^2 + a)^2 + b,$$ where $a$ and $b$ are constants - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 1

Step 1

Find the value of a and the value of b.

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Answer

To find the values of $a$ and $b$ , we can compare coefficients.
Starting with the equation:

$x^4 - 8x - 29 = (x^2 + a)^2 + b,$
we expand $(x^2 + a)^2$ to get:

$(x^2 + a)^2 = x^4 + 2ax^2 + a^2.$

Setting this equal to the left-hand side gives us:

$x^4 + 2ax^2 + a^2 + b = x^4 - 8x - 29.$

This leads to the following equations by comparing coefficients:

For $x^2$ :

ightarrow a = 0.$$

For constant terms:
$a^2 + b = -29.$
Substituting $a = 0$ gives:
$b = -29.$

Thus, the values of $a$ and $b$ are:

$a = 0$
$b = -29$ .

Step 2

Hence, or otherwise, show that the roots of x^4 - 8x - 29 = 0 are c ± √d, where c and d are integers to be found.

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Answer

Now substituting $a$ and $b$ , we have:

$(x^2 - 4)^2 = 45.$

Rearranging gives:

$(x^2 - 4)^2 - 45 = 0,$
which can be rewritten as:

$(x^2 - 4)^2 = 45.$

Taking the square root on both sides:

$x^2 - 4 = ext{±} rac{3}{2},$
thus we have:

$x^2 = 4 + rac{3}{2} = rac{11}{2}$ ;
$x^2 = 4 - rac{3}{2} = rac{5}{2}$ .

Taking square roots gives:

$x = \pm rac{ rac{11}{2} + rac{5}{2}}{2} = \pm \sqrt7,$
which indicates the values of $c$ and $d$ can be identified.
Thus, we have:

$c = 2$
$d = 5$ .

Given the equation: $$x^4 - 8x - 29 = (x^2 + a)^2 + b,$$ where $a$ and $b$ are constants - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 1

Worked Solution & Example Answer:Given the equation: $$x^4 - 8x - 29 = (x^2 + a)^2 + b,$$ where $a$ and $b$ are constants - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 1

Find the value of a and the value of b.

Hence, or otherwise, show that the roots of x^4 - 8x - 29 = 0 are c ± √d, where c and d are integers to be found.

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