(a) The value of the expression \(2Ax - Kx^{2}\) is 7 when x = 1 and 4 when x = 2. Find the values of the constants A and K. (b) Solve the equation \(x^{2} ...
Question 1 Report
(a) The value of the expression \(2Ax - Kx^{2}\) is 7 when x = 1 and 4 when x = 2. Find the values of the constants A and K.
(b) Solve the equation \(x^{2} - 3x - 1 = 0\), giving your answers correct to 1 decimal place.
Answer Details
(a) Substitute the two conditions into \(2Ax - Kx^2\).
At \(x = 1\): \(2A - K = 7\). \quad(1)
At \(x = 2\): \(4A - 4K = 4\Rightarrow A - K = 1\). \quad(2)
Subtract (2) from (1): \((2A - K) - (A - K) = 7 - 1\Rightarrow A = 6\). Then from (2), \(K = A - 1 = 5\).
\[A = 6,\qquad K = 5.\]
(b) Solve \(x^2 - 3x - 1 = 0\) using the quadratic formula:
\[x = \frac{-(-3)\pm\sqrt{(-3)^2 - 4(1)(-1)}}{2(1)} = \frac{3\pm\sqrt{13}}{2} = \frac{3\pm3.606}{2}.\]
\[x = 3.3\quad\text{or}\quad x = -0.3\ (\text{to 1 d.p.}).\]
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