Question

completing the square method 2x²+5x-3=0​

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

• We have been given a Quadratic Polynomial 2x² + 5x - 3 = 0

To Find:

• We have to find the roots of Equation using completing the square method

Completing Square Method:

Completing Square Method is used to solve a Quadratic equation by changing the equation so that the left side is a perfect square

Solution:

Given a Quadratic Polynomial

$\boxed{\sf{ 2 x^2 + 5x - 3 = 0}}$

Converting coefficient of x² to 1 by divding both sides of by 2

$\implies \sf{\dfrac{2}{2} x^2 + \dfrac{5}{2} x - \dfrac{3}{2} = \dfrac{0}{2}}$

$\implies \sf{x^2 + \dfrac{5}{2} x - \dfrac{3}{2} = 0}$

Shifting all coefficients to left side of the Equation

$\implies \sf{x^2 + \dfrac{5}{2} x = \dfrac{3}{2}}$

Dividing the coefficient of x by 2

• $\sf{\dfrac{Coefficient \: of \: x}{2}}$

• $\dfrac{5}{2} \times \dfrac{1}{2}$

• $\dfrac{5}{4}$

Adding ( 5/4 )² on Both sides of equation

$\implies \sf{x^2 + \left ( \dfrac{5}{4} \right )^2 + \dfrac{5}{2} x = \dfrac{3}{2} + \left ( \dfrac{5}{4} \right )^2}$

$\implies \sf{x^2 + \left ( \dfrac{5}{4} \right )^2 + 2 \times \dfrac{5}{4} \times x + = \dfrac{3}{2} + \dfrac{5}{4}}$

Using the below identity in above equation

$\boxed{\text{ ( a + b )² = a² + b² + 2ab }}$

$\implies \sf{\left ( x + \dfrac{5}{4} \right )^2 = \dfrac{3}{2} + \dfrac{25}{16}}$

Taking LCM on Right Hand side

$\implies \sf{\left ( x + \dfrac{5}{4} \right )^2 = \dfrac{3 \times 8 + 25}{16}}$

$\implies \sf{\left ( x + \dfrac{5}{4} \right )^2 = \dfrac{24 + 25}{16}}$

$\implies \sf{\left ( x + \dfrac{5}{4} \right )^2 = \dfrac{49}{16}}$

Taking Square root on Both sides

$\implies \sf{ x + \dfrac{5}{4} = \sqrt{\dfrac{49}{16} \: }}$

$\implies \sf{ x + \dfrac{5}{4} = \pm \: \dfrac{7}{4}}$

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$\odot \:$ Case 1 : Taking 7/4

$\implies \sf{ x + \dfrac{5}{4} = \dfrac{7}{4}}$

$\implies \sf{ x = \dfrac{7}{4} - \dfrac{5}{4}}$

$\implies \sf{ x = \dfrac{2}{4}}$

$\implies \boxed{\sf{ x = \dfrac{1}{2}} }$

$\sf{ }$

$\odot \:$ Case 2 : Taking - 7/4

$\implies \sf{ x + \dfrac{5}{4} = - \dfrac{7}{4}}$

$\implies \sf{ x = - \dfrac{7}{4} - \dfrac{5}{4} }$

$\implies \sf{ x = - \dfrac{12}{4} }$

$\implies \boxed{\sf{ x = - 3 }}$

Roots of Equation are 1/2 and - 3

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$\large\purple{\underline{\underline{\sf{Extra \: Information:}}}}$

• Roots of Equation are the value of x for which the value of given polynomial becomes zero
• Method of Finding roots
• Middle Term Splitting
• Completing Square Method
• Cross Multiplication Method