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:
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.
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:
.
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
Example
Solve 4x2 + x = 3 by completing the square.
Exercises
Solve the following quadratic equations by completing the square
Q1. 2s2 + 5s = 3
Q2. 3x2 = 3 − 4x
Q3. 9v2 − 6v − 2 = 0
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