O-Level Equations and inequalities: worked solution

Question

Use that result to solve x² + 6x + 7 = 0, rounding each root to two decimal places.

Part (b)(i) put the expression into completed-square form: x² + 6x + 7 = (x + 3)² − 9 + 7 = (x + 3)² − 2. Step 1 - use that result to rewrite the equation. x² + 6x + 7 = 0 becomes (x + 3)² − 2 = 0. Step 2 - make the squared bracket the subject. Add 2 to both sides: (x + 3)² = 2. Step 3 - take the square root of both sides. A square root gives TWO signs, not one, so x + 3 = √2 or x + 3 = −√2, usually written x + 3 = ±√2. Step 4 - subtract 3 from both sides. x = −3 ± √2, and √2 = 1.41421... Step 5 - evaluate each root and round to 2 d.p. x = −3 + 1.41421... = −1.58578... → −1.59 x = −3 − 1.41421... = −4.41421... → −4.41 So the roots are x ≈ −1.59 and x ≈ −4.41. (Check by the quadratic formula: x = (−6 ± √(36 − 28))/2 = (−6 ± √8)/2 = −3 ± √2, the same thing.)