(a) Given the expression \(y = ax^{2} - bx - 12\) , find the values

Explanation

(a) \(y = ax^{2} - bx - 12\)

When a = 1, b = 2 and y = 3.

\(3 = x^{2} - 2x - 12\)

\(x^{2} - 2x - 12 - 3 = 0 \implies x^{2} - 2x - 15 = 0\)

\(x^{2} - 5x + 3x - 15 = 0\)

\((x - 5)(x + 3) = 0\)

\(\text{x = 5 or -3}\)

(b) \(\sqrt{x^{2} + 1} = \frac{5}{4}\)

Squaring both sides,

\(x^{2} + 1 = \frac{25}{16}\)

\(x^{2} = \frac{25}{16} - 1 = \frac{9}{16}\)

\(x = \sqrt{\frac{9}{16}} = \pm \frac{3}{4}\)

The positive value of x = \(\frac{3}{4}\).