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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+18x+45.(x+15)(x+5)(x+15)(x+3)(x+14)(x+3) Q 2: Factorize, x2+10x+16.(x+8)(x+2)(x+7)(x+2)(x+8)(x+3) Q 3: Factorize, x2+14x+45.(x+8)(x+5)(x+9)(x+5)(x+9)(x+7) Q 4: Factorize, x2+22x+85.(x+15)(x+5)(x+17)(x+4)(x+17)(x+5) Q 5: Factorize, x2+30x+161.(x+23)(x+8)(x+21)(x+7)(x+23)(x+7) Q 6: Factorize, x2+16x+39.(x+13)(x+3)(x+13)(x+5)(x+14)(x+3) Question 7: This question is available to subscribers only! Question 8: This question is available to subscribers only!

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