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Middle/High School Algebra, Geometry, and Statistics (AGS)
4.1 Quadratic Equations - I

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.
Q 1: Factorize, x2+30x+161.
(x+21)(x+7)
(x+23)(x+7)
(x+23)(x+8)

Q 2: Factorize, x2+16x+39.
(x+14)(x+3)
(x+13)(x+3)
(x+13)(x+5)

Q 3: Factorize, x2+16x+48.
(x+11)(x+4)
(x+12)(x+6)
(x+12)(x+4)

Q 4: Factorize, x2+18x+65.
(x+12)(x+5)
(x+13)(x+4)
(x+13)(x+5)

Q 5: Factorize, x2+14x+45.
(x+9)(x+5)
(x+8)(x+5)
(x+9)(x+7)

Q 6: Factorize, x2+10x+16.
(x+7)(x+2)
(x+8)(x+3)
(x+8)(x+2)

Question 7: This question is available to subscribers only!

Question 8: This question is available to subscribers only!


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