f(x) = x² - 10x + 23 (a) Express f(x) in the form (x + a)² + b, where a and b are constants to be found - Edexcel - A-Level Maths Pure - Question 5 - 2018 - Paper 1

To find the exact solutions to $x^2 - 10x + 23 = 0$, we can use the completed square from part (a):

1. Set the completed square equal to zero:

   $(x - 5)^2 - 2 = 0$

2. Rearranging gives:

   $(x - 5)^2 = 2$

3. Taking the square root of both sides:

   $x - 5 = ext{±}\\sqrt{2}$

4. Hence, solving for x gives:

   $x = 5 ext{ ± } \\sqrt{2}$

Thus, the solutions are $x = 5 + \\sqrt{2}$ and $x = 5 - \\sqrt{2}$.

To find the larger solution to the equation $y - 10^{0.5} + 23 = 0$:

1. Solve for y:

   $y = 10^{0.5} - 23$

2. Since $10^{0.5} = \\sqrt{10}$, we substitute:

   $y = \\sqrt{10} - 23$

Thus, the solution can be expressed in the form $p + q\\sqrt{r}$, where:

• $p = -23$
• $q = 1$
• $r = 10$

Therefore, the final answer is:

$y = -23 + \\sqrt{10}$