1.5 Radical equations An equation involving expression under the radical sign is called a radical equation. e.g., x + 3 = x + 1 x + 3 = x + 1 \sqrt{x + 3} = x + 1 x + 3
= x + 1 x + 3
= x + 1 x − 1 = x − 2 + 1 x − 1 = x − 2 + 1 \sqrt{x - 1} = \sqrt{x - 2} + 1 x − 1
= x − 2
+ 1 x − 1
= x − 2
+ 1
1.5 (i) a x + b = c x + d a x + b = c x + d \sqrt{ax + b} = cx + d a x + b
= c x + d a x + b
= c x + d Solve the equation 3 x + 7 = 2 x + 3 3 x + 7 = 2 x + 3 \sqrt{3x + 7} = 2x + 3 3 x + 7
= 2 x + 3 3 x + 7
= 2 x + 3
3 x + 7 = 2 x + 3 3 x + 7 = 2 x + 3 \sqrt{3x + 7} = 2x + 3 3 x + 7
= 2 x + 3 3 x + 7
= 2 x + 3 (i)
Squaring both sides of the equation (i) , we get ( 3 x + 7 ) 2 = ( 2 x + 3 ) 2 ( 3 x + 7 ) 2 = ( 2 x + 3 ) 2 (\sqrt{3x + 7})^{2} = (2x + 3)^{2} ( 3 x + 7
) 2 = ( 2 x + 3 ) 2 ( 3 x + 7
) 2 = ( 2 x + 3 ) 2 3 x + 7 = 4 x 2 + 12 x + 9 3x + 7 = 4x^{2} + 12x + 9 3 x + 7 = 4 x 2 + 12 x + 9
Simplifying the above equation, we have 4 x 2 + 9 x + 2 = 0 4x^{2} + 9x + 2 = 0 4 x 2 + 9 x + 2 = 0
Applying quadratic formula, x = − 9 ± ( 9 ) 2 − 4 × 4 × 2 2 × 4 x = − 9 ± ( 9 ) 2 − 4 × 4 × 2 2 × 4 x = \frac{-9\pm\sqrt{(9)^{2} - 4\times4\times2}}{2\times4} x = 2 × 4 − 9 ± ( 9 ) 2 − 4 × 4 × 2
x = 2 × 4 − 9 ± ( 9 ) 2 − 4 × 4 × 2
= − 9 ± 81 − 32 8 = − 9 ± 49 8 = − 9 ± 7 8 = − 9 ± 81 − 32 8 = − 9 ± 49 8 = − 9 ± 7 8 = \frac{-9\pm\sqrt{81 - 32}}{8} = \frac{-9\pm\sqrt{49}}{8} = \frac{-9\pm7}{8} = 8 − 9 ± 81 − 32
= 8 − 9 ± 49
= 8 − 9 ± 7 = 8 − 9 ± 81 − 32
= 8 − 9 ± 49
= 8 − 9 ± 7 ∴ x = − 9 + 7 8 = − 2 8 = − 1 4 ∴ x = − 9 + 7 8 = − 2 8 = − 1 4 \therefore x = \frac{-9 + 7}{8} = \frac{-2}{8} = \frac{-1}{4} ∴ x = 8 − 9 + 7 = 8 − 2 = 4 − 1 ∴ x = 8 − 9 + 7 = 8 − 2 = 4 − 1 or x = − 9 − 7 8 = − 16 8 = − 2 or x = − 9 − 7 8 = − 16 8 = − 2 \text{or} x = \frac{-9 - 7}{8} = \frac{-16}{8} = -2 or x = 8 − 9 − 7 = 8 − 16 = − 2 or x = 8 − 9 − 7 = 8 − 16 = − 2
Putting x = − 1 4 x = − 1 4 x = - \frac{1}{4} x = − 4 1 x = − 4 1 (i) , we have 3 ( − 1 4 ) + 7 = 2 ( − 1 4 ) + 3 ⇒ − 3 + 28 4 = − 1 2 + 3 ⇒ 25 4 = 5 2 w h i c h i s t r u e . 3 ( − 1 4 ) + 7 = 2 ( − 1 4 ) + 3 ⇒ − 3 + 28 4 = − 1 2 + 3 ⇒ 25 4 = 5 2 w h i c h i s t r u e . \sqrt{3\left(-\frac{1}{4}\right) + 7} = 2\left(-\frac{1}{4}\right) + 3\Rightarrow \sqrt{\frac{-3 + 28}{4}} = -\frac{1}{2} +3\Rightarrow \sqrt{\frac{25}{4}} = \frac{5}{2}\mathrm{~which~is~true}. 3 ( − 4 1 ) + 7
= 2 ( − 4 1 ) + 3 ⇒ 4 − 3 + 28
= − 2 1 + 3 ⇒ 4 25
= 2 5 which is true . 3 ( − 4 1 ) + 7
= 2 ( − 4 1 ) + 3 ⇒ 4 − 3 + 28
= − 2 1 + 3 ⇒ 4 25
= 2 5 which is true .
Putting x = − 2 x = - 2 x = − 2 (i) , we have 3 ( − 2 ) + 7 = 2 ( − 2 ) + 3 3 ( − 2 ) + 7 = 2 ( − 2 ) + 3 \sqrt{3\left(-2\right) + 7} = 2\left(-2\right) + 3 3 ( − 2 ) + 7
= 2 ( − 2 ) + 3 3 ( − 2 ) + 7
= 2 ( − 2 ) + 3 ⇒ − 6 + 7 = − 4 + 3 ⇒ 1 = − 1 ⇒ − 6 + 7 = − 4 + 3 ⇒ 1 = − 1 \Rightarrow \sqrt{-6 + 7} = -4 +3\Rightarrow \sqrt{1} = -1 ⇒ − 6 + 7
= − 4 + 3 ⇒ 1
= − 1 ⇒ − 6 + 7
= − 4 + 3 ⇒ 1
= − 1 x = − 2 x = -2 x = − 2 { − 1 4 } { − 1 4 } \left\{-\frac{1}{4}\right\} { − 4 1 } { − 4 1 }