Question

Solve: (x – 4)(x – 2)(x- 7)(x + 1) = 16

Answer 1

(x – 4)(x – 7) (x – 2)(x + 1) = 16

⇒ (x – 4)(x – 2)(x – 7)(x + 1) = 16

⇒ (x² – 6x + 8)(x² – 6x – 7) = 16

Let x² – 6x = y

(y + 8)(y – 7) = 16

⇒ y² – 7y + 8y – 56 – 16 = 0

⇒ y² + y – 72 = 0

⇒ (y + 9)(y – 8) = 0

y + 9 = 0

x² – 6x + 9 = 0

(x – 3)² = 0

x = 3, 3

or

y – 8 = 0

x² – 6x – 8 = 0

x = ${\displaystyle \frac{6\pm \sqrt{36-4\left(1\right)(-8)}}{2\left(1\right)}}$

= ${\displaystyle \frac{6\pm \sqrt{36+32}}{2}}$

= ${\displaystyle \frac{6+\sqrt{68}}{2}}$

= ${\displaystyle \frac{6\pm 2\sqrt{17}}{2}}$

= ${\displaystyle 2\frac{(3\pm \sqrt{17})}{2}}$

= ${\displaystyle 3+\sqrt{17}}$

Roots are ${\displaystyle 3,3,3\pm \sqrt{17}}$