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Solution of a Quadratic Equation by Completing the Square

Steps for completing the square

Step 1
Write an equivalent equation with only the x2 term and the x term on the left side of the equation. The coefficient of the x2 term must be 1.

Step 2
Add the square of one-half the coefficient of the x term to both sides of the equation.

Step 3
Express the left side of the equation as a perfect square.

Step 4
Solve for x.

For any quadratic equation in the form ax2 + bx + c = 0, where a 0, the two solutions are

We will prove the quadratic theorem by completing the square on ax2 + bx + c = 0.

ax2 + bx + c = 0

This completes the proof. Let us see what we have proved. If our equation is in the form ax2 + bx + c = 0 (standard form), where a 0, the two solutions are always given by the formula

This formula is known as the quadratic formula. If we substitute the coefficients a, b and c of any quadratic equation in standard form in the formula, we need to perform some basic arithmetic to arrive at the solution set. 


This method was developed by the noted mathematician Sridharcharya. If b2 - 4ac 0, then quadratic equation ax2 + bx + c = 0 has two real roots
α and β which will be given by,

α = and β =

The quantity 'b2 - 4ac' is called, discriminant of the equation ax2 + bx + c = 0 and is denoted by the letter 'D'.
Therefore D = b2 - 4ac.



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