Photo AI
15 (a) Show that the equation $x^2 + 7x - 5 = 0$ has a solution between $x = 0$ and $x = 1$ - Edexcel - GCSE Maths - Question 15 - 2017 - Paper 3
Question 15
15 (a) Show that the equation $x^2 + 7x - 5 = 0$ has a solution between $x = 0$ and $x = 1$. (b) Show that the equation $x^2 + 7x - 5 = 0$ can be arranged to give $... show full transcript
Worked Solution & Example Answer:15 (a) Show that the equation $x^2 + 7x - 5 = 0$ has a solution between $x = 0$ and $x = 1$ - Edexcel - GCSE Maths - Question 15 - 2017 - Paper 3
Step 1
Show that the equation $x^2 + 7x - 5 = 0$ has a solution between $x = 0$ and $x = 1$.
Answer
To show that the equation has a solution in this interval, we evaluate the function at the endpoints:
Let ( f(x) = x^2 + 7x - 5 ).
-
Calculating ( f(0) ): [ f(0) = 0^2 + 7(0) - 5 = -5 ]
-
Calculating ( f(1) ): [ f(1) = 1^2 + 7(1) - 5 = 1 + 7 - 5 = 3 ]
Since ( f(0) = -5 ) and ( f(1) = 3 ), there is a sign change between these two values (from negative to positive). By the Intermediate Value Theorem, there is at least one root in the interval ((0, 1)).
Step 2
Show that the equation $x^2 + 7x - 5 = 0$ can be arranged to give $x = \frac{5}{x + 7}.$
Answer
Starting from the equation ( x^2 + 7x - 5 = 0 ), we can rearrange it as follows:
-
Add 5 to both sides: [ x^2 + 7x = 5 ]
-
Isolate ( x ): [ x = \frac{5 - 7x}{x} ] This simplifies to: [ x = \frac{5}{x + 7} ]
Thus, we have shown the required rearrangement.
Step 3
Starting with $x_1 = 1$, use the iteration formula $x_{n + 1} = \frac{5}{x_n + 7}$ three times to find an estimate for the solution of $x^2 + 7x - 5 = 0$.
Answer
Using the iteration formula:
-
Start with ( x_1 = 1 ): [ x_2 = \frac{5}{1 + 7} = \frac{5}{8} = 0.625 ]
-
Next iteration: [ x_3 = \frac{5}{0.625 + 7} = \frac{5}{7.625} \approx 0.6564 ]
-
Last iteration: [ x_4 = \frac{5}{0.6564 + 7} \approx \frac{5}{7.6564} \approx 0.6541 ]
Thus, after three iterations, we estimate that the solution is approximately ( x \approx 0.6541 ).
Step 4
By substituting your answer to part (c) into $x^2 + 7x - 5$, comment on the accuracy of your estimate for the solution to $x^2 + 7x - 5 = 0$.
Answer
Substituting ( x \approx 0.6541 ) into the equation:
- Calculate: [ f(0.6541) = (0.6541)^2 + 7(0.6541) - 5 ] [ \approx 0.4288 + 4.5787 - 5 \approx 0.0075 ]
Since ( f(0.6541) \approx 0.0075 ), which is very close to zero, we can conclude that our estimate is quite accurate, suggesting that the solution to the equation lies close to ( 0.6541 ).