GCSE Edexcel Maths Iteration: Using $x_{int} = -2

Using $x_{int} = -2 - \frac{4}{x_{int}}$ with $x_{int} = -2.5$ (a) find the values of $x_1$, $x_2$, and $x_3$ - Edexcel - GCSE Maths - Question 16 - 2017 - Paper 3

Question 16

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Using $x_{int} = -2 - \frac{4}{x_{int}}$ with $x_{int} = -2.5$ (a) find the values of $x_1$, $x_2$, and $x_3$. (b) Explain the relationship between the values of... show full transcript

Worked Solution & Example Answer:Using $x_{int} = -2 - \frac{4}{x_{int}}$ with $x_{int} = -2.5$ (a) find the values of $x_1$, $x_2$, and $x_3$ - Edexcel - GCSE Maths - Question 16 - 2017 - Paper 3

Step 1

find the values of $x_1$, $x_2$, and $x_3$

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Answer

To find the values, we start by substituting $x_{int} = -2.5$ into the equation:

$x_{int} = -2 - \frac{4}{x_{int}}$

This leads to:

$-2.5 = -2 - \frac{4}{-2.5}$

Next, we calculate:

$-2 - \frac{4}{-2.5} = -2 + 1.6 = -0.4$

Thus:

$x_1 = -0.4$ .

Now, we will continue this process for subsequent iterations:

Substitute $x_1 = -0.4$ into the equation: $x_2 = -2 - \frac{4}{x_1} = -2 - \frac{4}{-0.4} = -2 + 10 = 8$ Therefore: $x_2 = 8$ .
Finally, substitute $x_2 = 8$ : $x_3 = -2 - \frac{4}{x_2} = -2 - \frac{4}{8} = -2 - 0.5 = -2.5$ Thus: $x_3 = -2.5$ .

Step 2

Explain the relationship between the values of $x_1$, $x_2$, and $x_3$, and the equation $x^2 + 2x + 4 = 0$

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Answer

The values of $x_1$ , $x_2$ , and $x_3$ are iterations that reflect the process of estimating solutions for the equation $x^2 + 2x + 4 = 0$ . The iterations converge upon certain values that represent estimates of the roots.

The equation $x^2 + 2x + 4 = 0$ can be analyzed using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a=1$ , $b=2$ , and $c=4$ . The discriminant is:

$D = b^2 - 4ac = 2^2 - 4*1*4 = 4 - 16 = -12$

Since the discriminant is negative, the equation has no real solutions and the estimated values from the iterations provide a means of converging to complex solutions.

Using $x_{int} = -2 - \frac{4}{x_{int}}$ with $x_{int} = -2.5$ (a) find the values of $x_1$, $x_2$, and $x_3$ - Edexcel - GCSE Maths - Question 16 - 2017 - Paper 3

Worked Solution & Example Answer:Using $x_{int} = -2 - \frac{4}{x_{int}}$ with $x_{int} = -2.5$ (a) find the values of $x_1$, $x_2$, and $x_3$ - Edexcel - GCSE Maths - Question 16 - 2017 - Paper 3

find the values of $x_1$, $x_2$, and $x_3$

Explain the relationship between the values of $x_1$, $x_2$, and $x_3$, and the equation $x^2 + 2x + 4 = 0$

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