Question

Find a quadratic equation having roots -2 and -6.

Answer 1

Hint: Use factor theorem which states that if a polynomial p(x) vanishes at x = a, then x-a is a factor of p(x). Also, the number of linear factors in which a quadratic polynomial can be factored is 2. Using this information, find a general form of the quadratic polynomials with roots -2 and -6. Substitute a particular value or the variable and hence find a quadratic equation
Alternatively, use the fact that if a and b are the roots of a quadratic equation, then one such equation is given by ${{x}^{2}}-\left( a+b \right)x+ab=0$.

Complete step-by-step answer:
We have since -2 is a root of the quadratic equation
Hence x+2 is a factor of the quadratic expression
Also since -6 is a root of the quadratic equation, we have
x+2 is a factor of the quadratic expression
Hence the expression is of the form A(x+2)(x+6), where A is an arbitrary non-zero real number.
Put A = 1; we get the quadratic expression is given by
(x+2)(x+6) .
Hence the quadratic equation is
(x+2)(x+6) = 0
Using $\left( x+a \right)\left( x+b \right)={{x}^{2}}+\left( a+b \right)x+ab$, we get
${{x}^{2}}+8x+12=0$ is a quadratic equation with roots -2 and -6.

Note: We know that if a and b are the roots of a quadratic equation, then one such equation is given by ${{x}^{2}}-\left( a+b \right)x+ab=0$.
Put a = -2 and b = -6, we get
The quadratic expression with roots -2 and -6 is ${{x}^{2}}-\left( -2-6 \right)x+\left( -2 \right)\left( -6 \right)=0$
Hence the quadratic equation with roots -2 and -6 is ${{x}^{2}}+8x+12=0$.