GRE Algebra | Solving Quadratic Equations

In algebra, a quadratic equation can be written in the form:

ax2 + bx + c = 0 

where x is the variable and a, b, c are the real numbers and a≠0. If a=0 then it will be a linear equation not quadratic because no second order term. If quadratic equation has solution then it can be found by using the quadratic formula.

x = \frac{-b\pm \sqrt{b^2-4ac}}{2a} 

• Example-1: Solve the quadratic equation for x,

  x2 + 10x -24 = 0 

  Solution: In the quadratic equation, we have,

  a=1, b=10 and c=-24 

  Therefore the quadratic formula yields

  x = \frac{-10\pm \sqrt{10^2-4(1)(-24)}}{2(1)} 
  
  x = \frac{-10\pm \sqrt{100+96}}{2} 
  
  x = \frac{-10\pm 14}{2}  

  Hence, two solutions for the above equations are:

  x = 4/2 = 2,
  And
  x = -24/2 = -12 

• Example-2: Solve the quadratic equation using factorization,

  x2 + 2x - 15 = 0

  Solution: Given equation,

  x2 + 2x - 15 = 0 

  It can be factorize as,

  x2 -3x + 5x - 15 = 0
  x(x - 3) + 5(x - 3) = 0
  (x - 3)(x + 5) = 0 

  Hence, two solutions for x are:

  x = 3 and x = -5