2. Solving Quadratic Equations by Completing the Square


For quadratic equations that cannot be solved by factorising, we use a method which can solve ALL quadratic equations called completing the square. We use this later when studying circles in plane analytic geometry.

Completing the square comes from considering the special formulas:

math expression

Follow these steps for the quadratic equation

ax2 + bx + c = 0

(i) If a does not equal 1, divide each side by a (so that the coefficient of the x2 is 1).

(ii) Rewrite the equation with the constant term on the right side.

(iii) Complete the square by adding the square of one-half of the coefficient of x to both sides.

(iv) Write the left side as a square and simplify the right side.

(v) Equate and solve.

 

Example:

Find the roots of x2 + 10x − 4 = 0 using completing the square method.


Let's see the LiveMath solution first.

LIVEMath

 

Answer

Step (i) a = 1 [no action necessary]

Step (ii) Rewrite the equation with the constant term on the right side.

x2 + 10x = 4

Step (iii) Complete the square by adding the square of one-half of the coefficient of x to both sides. In this case:

math expression.

x2 + 10x + 25 = 4 + 25

x2 + 10x + 25 = 29

Step (iv) Write the left side as a square:

(x + 5)2 = 29

Step (v) Equate and solve

math expression

 

Example

Solve 4x2 + x = 3 by completing the square.


Answer

Exercises

Solve the following quadratic equations by completing the square

Q1. 2s2 + 5s = 3

Q2. 3x2 = 3 − 4x

Q3. 9v2 − 6v − 2 = 0


Answer



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