Question

If x^2+ax+b is a perfect square what is the relation between a and b

Answer 1

The required relation is a² = 4b

Step-by-step explanation:

The given expression is (x² + ax + b)

Now, x² + ax + b

= x² + 2 * x * a/2 + (a/2)² + b - (a/2)²

= (x + a/2)² + {b - (a/2)²}

To obtain a perfect square, we must have the non-squared term zero.

Then,

b - (a/2)² = 0

or, b - a²/4 = 0

or, a²/4 = b

or, a² = 4b

Therefore the required relation is a² = 4b.

Note: To solve this type of problems, first find where the x² and x terms are. Then use algebraic operations to get a perfect square term and you will also get a constant part, containing arbitrary constants. Then just equate the constant part with zero.

Answer 2

If $x^{2}+ax+b$ is a perfect square, then the relation between a and b is given by $a^{2}=4b$.

• A quadratic polynomial of the form $ax^{2}+bx+c$ is said to be a perfect square if it has two equal roots.
• In this case, discriminant is equal to zero.

                      ⇒ $b^{2}-4ac=0$

• From the given equation,

                     a = 1, b = a, c = b

                      ⇒ $a^{2}-4(1)(b)=0$

                      ⇒ $a^{2}=4b$