Theory question 107
Quadratic Equations(a) Using completing the squares method, solve, correct to 2 decimal places, (x − 2)/4 = (x + 2)/2x.
(b) In the diagram, PQRST is a circle with centre O. If PS is a diameter, RS//QT, |QR| = |RS| and ∠QTS = 52°, find: (i) ∠SQT; (ii) ∠PQT.
Show the model answer
Model answer
(a) 2x(x − 2) = 4(x + 2) → 2x² − 8x − 8 = 0 → x² − 4x = 4
x² − 4x + 4 = 8 → (x − 2)² = 8 → x − 2 = ±2.828
x ≈ 4.83 or x ≈ −0.83
(b)(i) ∠QRS = 180° − 52° = 128° (opposite angles of cyclic quadrilateral QRST)
Triangle QRS is isosceles: ∠RSQ = (180° − 128°)/2 = 26°
∠SQT = ∠RSQ = 26° (alternate angles, RS//QT)
(ii) ∠PQS = 90° (angle in a semicircle), so ∠PQT = 90° − 26° = 64°
Explanation
Add the square of half the coefficient of x to both sides.