Completing the square

method of completing the square

Completing the square is a mathematical approach to solve any quadratic equation for obtaining the roots. In this method, a quadratic equation is transformed into a square of a binomial and then roots are the quadratic equation are evaluated.

For example, $ax^2+bx+c = 0$ is a quadratic equation and this equation is converted as a square of a binomial and the value of the constant is shifted to right hand side of the equation as follows.

${(x+p)}^2 = q$

The following example understands you the method of completing the square to solve any quadratic equation in mathematics.

Example

Completing the square method is a standard mathematical approach and any quadratic equation can be solved in five simple steps.

$2x^2 -7x + 3 = 0$ is an example quadratic equation to understand the procedure of completing the square method.

$Step: 1$
Move the constant term to right hand side of the equation.

$\implies 2x^2 -7x = -3$

$Step: 2$
Divide both sides of the equation by the coefficient of $x^2$.

$\implies \dfrac{2x^2 -7x}{2} = -\dfrac{3}{2}$

$\implies \dfrac{2x^2}{2} -\dfrac{7x}{2} = -\dfrac{3}{2}$

$\implies \Bigg(\dfrac{2}{2}\Bigg)x^2 -\Bigg(\dfrac{7}{2}\Bigg)x = -\dfrac{3}{2}$

$\implies \require{cancel} \Bigg(\dfrac{\cancel{2}}{\cancel{2}}\Bigg)x^2 -\Bigg(\dfrac{7}{2}\Bigg)x = -\dfrac{3}{2}$

$\implies x^2 -\Bigg(\dfrac{7}{2}\Bigg)x = -\dfrac{3}{2}$

$\implies x^2 -x\Bigg(\dfrac{7}{2}\Bigg) = -\dfrac{3}{2}$

$Step: 3$
Transform the left hand side expression as a complete square of a binomial.

$\implies x^2 -\Bigg(\dfrac{2}{2}\Bigg)x\Bigg(\dfrac{7}{2}\Bigg) = -\dfrac{3}{2}$

$\implies x^2 -2x\Bigg(\dfrac{7}{4}\Bigg) = -\dfrac{3}{2}$

$\implies x^2 -2x\Bigg(\dfrac{7}{4}\Bigg) + {\Bigg(\dfrac{7}{4}\Bigg)}^2 -{\Bigg(\dfrac{7}{4}\Bigg)}^2 = -\dfrac{3}{2}$

$\implies x^2 -2x\Bigg(\dfrac{7}{4}\Bigg) + {\Bigg(\dfrac{7}{4}\Bigg)}^2 -\dfrac{49}{16} = -\dfrac{3}{2}$

$\implies x^2 -2x\Bigg(\dfrac{7}{4}\Bigg) + {\Bigg(\dfrac{7}{4}\Bigg)}^2 = \dfrac{49}{16} -\dfrac{3}{2}$

$\implies x^2 -2x\Bigg(\dfrac{7}{4}\Bigg) + {\Bigg(\dfrac{7}{4}\Bigg)}^2 = \dfrac{49 -24}{16}$

$\implies {\Bigg(x -\dfrac{7}{4}\Bigg)}^2 = \dfrac{25}{16}$

$Step: 4$
Square root both sides of the equation.

$\implies \Bigg(x -\dfrac{7}{4}\Bigg) = \pm \sqrt{\dfrac{25}{16}}$

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

$Step: 5$
Evaluate the roots of the Quadratic equation by finding the values of $x$.

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

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

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

$\implies x = \dfrac{12}{4}$ and $x = \dfrac{2}{4}$

$\therefore \,\,\, x = 3$ and $x = \dfrac{1}{2}$

Therefore, the roots of quadratic equation $2x^2 -7x + 3 = 0$ by using completing the square method are $3$ and $\dfrac{1}{2}$.

Now test your skill with our worksheet and practice the completing the square method to solve quadratic equations and obtain their solutions.


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