Question

Find the roots of the quadratic equation
$2x {}^{2} - 17x + 21 = 0$
By using factorisation, completing squares and quadratic formula ​

Answer 1

GIVEN :

Quadratic equation : 2x² - 17x + 21 = 0

TO FIND :

Roots of Equation using completing Square method and Quadratic formula.

SOLUTION :

Method : Completing Square

2x² - 17x + 21 = 0

Divide both sides with 2.

$\frac{ {2x}^{2} }{2} - \frac{17x}{2} + \frac{21}{2} = 0 \\ \\ {x}^{2} - \frac{17x}{2} = - \frac{21}{2}$

Add (17/4)² on both sides

${x}^{2} - \frac{17x}{2} + {( \frac{17}{4} )}^{2} = - \frac{21}{2} + {( \frac{17}{4} )}^{2} \\ \\ {(x - \frac{17}{4}) }^{2} = - \frac{21}{2} + \frac{289}{16} \\ \\ {(x - \frac{17}{4}) }^{2} = \frac{121}{16} \\ \\ x - \frac{17}{4} = \sqrt{ \frac{121}{16} } \\ \\ x - \frac{17}{4} = \frac{11}{4} \\ \\ x = \frac{11}{4} + \frac{17}{4} \: \: x = - \frac{11}{4} + \frac{17}{4} \\ \\ x = \frac{28}{4} \: \: \: \: \: x = \frac{6}{4} \\ \\ x = 7 \: \: x = \frac{3}{2}$

Therefore, zeroes are 7 and 3/2.

Method : Quadratic formula

2x² - 17x + 21

We know that,

Quadratic Formula :

$x = \frac{ - b \: + \: or \: - \: \sqrt{ {b}^{2} - 4ac } }{2a}$

From the Quadratic equation,

a = 2

b = -17

c = 21

Substitute the values in the formula.

$x = \frac{ - ( - 17) + \sqrt{ {( - 17)}^{2} - 4(2)(21)} }{4} \: \: x = \frac{ - ( - 17) - \sqrt{ {( - 17)}^{2} - 4(2)(21)} }{4} \\ \\ x = \frac{17 + \sqrt{289 - 168} }{4} \: \: x = \frac{17 - \sqrt{289 - 168} }{4} \\ \\ x = \frac{17 + \sqrt{121} }{4} \: \: x = \frac{17 - \sqrt{121} }{4} \\ \\ x = \frac{17 + 11}{4} \: \: x = \frac{17 - 11}{4} \\ \\ x = \frac{28}{4} \: \: x = \frac{6}{4} \\ \\ x = 7 \: \: \: x = \frac{3}{2}$

Therefore, the roots are 7 and 3/2.

Method : Factorisation

2x² - 17 + 21 = 0

Split the middle term

2x² - 14x - 3x +21= 0

2x(x - 7) - 3(x - 7) = 0

x - 7 = 0 2x - 3 = 0

x = 7 and x = 3/2.

Therefore, the roots are 7 and 3/2.