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
Question 5
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.
(b) Hence, or otherwise, find the exact solutions to the e... show full transcript
Worked Solution & Example Answer: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
Step 1
Express f(x) in the form (x + a)² + b
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Answer
To express the function f(x) = x² - 10x + 23 in the desired format, we start by completing the square.
Identify the coefficient of x, which is -10. Divide this by 2 to get -5 and then square it to get 25.
Rewrite f(x) as:
f(x)=(x2−10x+25)−25+23
This simplifies to:
f(x)=(x−5)2−2
Hence, we have a = -5 and b = -2.
Step 2
Hence, or otherwise, find the exact solutions to the equation x² - 10x + 23 = 0
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Answer
Using the completed square from part (a):
(x−5)2−2=0
Rearranging gives:
(x−5)2=2
Taking the square root of both sides results in:
x−5=ext±2
Therefore, the exact solutions are:
x=5ext±sqrt2.
Step 3
Use your answer to part (b) to find the larger solution to the equation y - 10^{0.5} + 23 = 0
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Answer
To solve the equation y - 10^{0.5} + 23 = 0:
Substitute the expression for y:
y=100.5−23
Plug in the value of the larger root from part (b):
=27+102
Thus, the final solution in the required form is:
p+qr
where p = 27, q = 10, and r = 2.
