Question

How do you factor \[2{x^2} + 22x + 60\]?

Answer 1

Hint:The problem deals with the quadratic equation. A polynomial having degree 2 in one variable in form of $ f\left( x \right) = a{x^2} + bx + c $ where $ a,b,c \in R $ and $ a \ne 0 $, when equated to zero becomes quadratic equation of degree 2.the general form of quadratic equation is $ f\left( x \right) = a{x^2} + bx + c $. $ a $ is called as the leading coefficient and $ c $ is called absolute term of quadratic equation. The value satisfying the quadratic equation is called the roots of the quadratic equation. The quadratic equation will always have two roots. The nature of the root may be either real or imaginary.
Formula for finding the root of the quadratic equation is $ \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $.
A quadratic equation having more than two roots or solutions either real or complex is known as identity.
The rule of signs for quadratic polynomial states that the number of positive real roots of $ {P_n}\left( x \right) = 0 $ will not be more than the number of sign changes and the number of negative roots will not be more than the number of sign changes in polynomial $ {P_n}\left( { - x} \right) = 0 $.

Complete step by step solution:
The given quadratic equation is \[2{x^2} + 22x + 60 = 0\].
Compare the given polynomial with the general form of the quadratic equation $ f\left( x \right) = a{x^2} + bx + c $.
Therefore, $ a = 2, $ $ b = 22 $ and $ c = 60 $.
Suppose the roots of the quadratic equation are $ \left( {\alpha ,\beta } \right) $.
Substitute the value of $ a,b, $ and $ c $ in the formula.
 $
  x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \\
   \Rightarrow x = \dfrac{{ - 22 \pm \sqrt {{{\left( {22} \right)}^2} - 4 \times 1 \times 60} }}{{2 \times 2}} \\
   \Rightarrow x = \dfrac{{ - 22 \pm \sqrt {484 - 480} }}{4} \\
  $
Solve the equation to find the value of $ x $.
 $
  x = \dfrac{{ - 22 \pm \sqrt 4 }}{4} \\
   \Rightarrow x = \dfrac{{ - 22 \pm 2}}{4} \\
  $
Consider the positive value of $ x $.
  $
  x = \dfrac{{ - 22 + 2}}{4} \\
   \Rightarrow x = \dfrac{{ - 20}}{4} \\
   \Rightarrow x = - 5 \\
  $
Consider the negative value of $ x $.
 $
  x = \dfrac{{ - 22 - 2}}{4} \\
   \Rightarrow x = \dfrac{{ - 24}}{4} \\
   \Rightarrow x = - 6 \\
  $
Therefore the roots of the quadratic equation are -5 and -6.

Note:
While solving the problems students are advised to use the formula of quadratic equation $ \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $. Compare the value of given quadratic equation with general equation of quadratic equation $ f\left( x \right) = a{x^2} + bx + c $ to find the coefficient of leading term and absolute term.