Given the expression: $$x^4 - 8x - 29 - (x^4 + a) + b,$$ 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 $c eq rac{ ext{D}}{4}$, where c and d are integers to be found.

A-Level Edexcel Maths Pure Simultaneous Equations: Given the expression: $$x^4

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

Question 5

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

Worked Solution & Example Answer:Given the expression: $$x^4 - 8x - 29 - (x^4 + a) + b,$$ 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 the coefficients in the given expression. Start with:

$x^4 - 8x - 29 - (x^4 + a) + b = 0$

This simplifies to:

$-8x - 29 - a + b = 0$

Setting coefficients equal to zero gives the following equations:

For the x-term:

$-8 = 0$ , This does not require a value for a since it cancels out after removing $x^4$ .
For the constant terms:
$b = a + 29$$$

Using the normalized polynomial comparison, we know that:

From the $x^4$ $x^{4}$ terms, the constant term’s mismatch leads to:
- Comparing the constant terms gives:
- If $- 45 = b$ , then
- Setting up the equations results:
b = -45$$.

Thus, we conclude:

The value of a is 4.
The value of b is -45.

Step 2

Hence, or otherwise, show that the roots of x^4 - 8x - 29 = 0 are c ± √d.

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Answer

To show that the roots of the equation

$x^4 - 8x - 29 = 0$

can be expressed in the form $c ± √d$ , we start by applying the quadratic formula. First, rewrite the equation as:

$x^4 - 8x - 29 = 0$

This can be factored or solved using numerical methods.

Reorganizing gives:

$x^4 - 8x = 29$
Substituting for $x^2$ allows us to extract roots:

$y^2 - 8y - 29 = 0$ where $y = x^2$ .
Employing the quadratic formula:

$y = rac{-(-8) ± ext{√}((-8)^2 - 4 imes 1 imes (-29))}{2 imes 1}$ This simplifies to:

$y = rac{8 \pm ext{\sqrt}(64 + 116)}{2} = rac{8 \pm ext{\sqrt}(180)}{2} = rac{8 \pm 6\sqrt5}{2}$ .
Thus, we find:

$y = 4 ± 3√5$
Since $y = x^2$ , we take square roots:

$x = ±√(4 ± 3√5)$

From this, we conclude that there exist integers constants c and d such that:

$x = c ± √d$
Specifically, we find c = 4 and d as the minimum degree bound of 5.

Hence, we have shown the desired form.

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

Worked Solution & Example Answer:Given the expression: $$x^4 - 8x - 29 - (x^4 + a) + b,$$ 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.

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