Definition:
- Polynomials of degree two are called quadratic experssions/equations.
- Quadratic equations are of the form ax2 + bx + c = 0. Where a,b,c are constants and a <> 0, is in standard form for a quadratic equation.
- The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term or constant term.
- Quadratic equations of type ax2 + bx + c = 0 and ax2 + bx = 0 (c is 0) can be factored to solve for x.
- If a = 0 then it become a linear equation.
- If the coefficient of second degree term is one (a=1) then the quadratic equation it is called a monic quadratic polynomial.
Example: x2+4x=7, x2-6x+3,............etc, are called
monic quadratic expression.
Solution of Monic Quadratic Equations:
Example:
Factorize, x2+7x+12.
Solution:
Given that, x2+7x+12.
A quadratic monic polynomial in x is a perfect square if
the constant term in it is equal to the square of half the coefficient of x.
Therefore, if (12/2)2 = (6)2 = 36 is added to x2+12x
it becomes a perfect square.
Therefore, we add and subtract 36 to given equation.
= x2+7x+12 = (x2+7x+36)-36+27
(x2+7x+36) this is form of (a2+2ab+b2) =
(a+b)2.
Here a = x and b = 6, substitute these values in the formula, we get
= (x+6)2-36+27
= (x+6)2-9
=(x+6)2-32
This is in the form of a2-b2 = (a+b)(a-b)
Here a = x+6 and b = 3, substitute these values in the formula, we get
= (x+6+3)(x+6-3)
= (x+9)(x+3)
Directions: Solve the following problems. Also write at least ten examples of your own.
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