## Problem Solving: Quadratic Equation

7/3/2014

Algebra is one of the main branches of mathematics and is concerned with the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers, variables, and polynomials, along with their factorization and determining their roots. It serves as a pre-requisite for higher-level courses in mathematics like Calculus. Algebra develops one's thinking, specifically logic, patterns, problem solving, and deductive and inductive reasoning.

Algebra is taught in schools by solving basic algebra problems and
then moving to a more advanced level course: Algebra 2. This article
presents a free algebra problem solver for solving a quadratic equation of the form
ax^{2}+bx+c. The solution of equation is derived using three methods: factoring and completing
the square and quadratic formula.

Solving the quadratic equation 2x^{2}+17x+21=0

**Factoring**

· Factor 2x^{2}+17x+21

· Multiply 2 by 21, which is 42.

· Find factors of 42 which is 2x7x3

· Ask: Which two numbers add up to 17 and multiply to 42?

· Answer: 14 and 3

· Rewrite 17x as the sum of 14x and 3x

2x^{2}+14x+3x+21=0

· Factor out common terms in the first two terms, then in the last two terms [2x(x+7)+3(x+7)]=0

· Factor out the common term x+7

(x+7)(2x+3)=0

· The solutions are x=-7 or x=-2/3

**Completing the square**

· Dividing the equation 2x^{2}+17x+21
by 2

x^{2}+17/2x+21/2=0

· Subtract -21/2 from both sides

x^{2}+17/2x=-21/2

· Complete the square by adding
(17/4)^{2}

x^{2}+17/2x+(17/4)^{2} =-21/2+(17/4)^{2} => (x+17/4)^{2}=-21/2+289/16 =>(x+17/4)^{2}=121/16

· Solve for x

x+17/4=±√(121/16)=±11/4 => x=11/4-17/4 or x=-11/4-17/4 =>x=-6/4=-2/3 or x=-28/4=-7

· The solutions are x=-7 or x=-2/3

** **

**Quadratic Formula**

The solutions of equation ax^{2}+bx+c where a≠0 can be found using the quadratic formula *x*=-*b*±*b*2-4*ac*2*a*

· Solve for x for 2x2+17x+21=0

· Values of a=2, b=17, c=21

· Solutions of x are

x=-17±√(17)2-4x2x21/2x2 = -17 ±√121/4 = -17± 11/4

· Either x=-28/4=-7 or x=-6/4=-2/3

The solutions are x=-7 or x=-2/3

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