Question

The difference between the roots of \[{x^2} - 13x + k = 0\] is 7, find k.

Answer 1

Hint: A polynomial is an expression which contains the variables and coefficients which involves operations like addition, subtraction, multiplication. Quadratic equations are the equation that contains at least one squared variable, which is equal to zero. Quadratic equations are useful in our daily life; they are used to calculate areas, speed of the objects, projection, etc.
 The roots of the equation are given by the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], where \[{b^2} - 4ac\] tells the nature of the solution.
Here, in the question, we need to determine the value of k in the quadratic expression \[{x^2} - 13x + k = 0\] such that the difference between the roots is 7. For this, we need to use the two properties of the quadratic equation, which are \[\alpha + \beta = \dfrac{{ - b}}{a};\alpha \beta = \dfrac{c}{a}\].

Complete step-by-step answer:
Given the quadratic equation is \[{x^2} - 13x + k = 0\]
Let \[\alpha \]and \[\beta \] be the roots of the quadratic equation, where
\[\alpha - \beta = 7 - - (i)\]
Now compare the given equation with the general quadratic equation; hence we can write
\[
\Rightarrow a = 1 \\
\Rightarrow b = - 13 \\
\Rightarrow c = k \\
 \]
We know the sum of the roots of a quadratic equation is
\[\alpha + \beta = - \dfrac{b}{a} = 13 - - (ii)\]
Now solve the equation (i) and (ii), by adding we get
\[
\Rightarrow \alpha - \beta + \alpha + \beta = 7 + 13 \\
  2\alpha = 20 \\
  \alpha = 10 \\
 \]
Now put the value of in equation (i) as:
\[
\Rightarrow \alpha - \beta = 7 \\
\Rightarrow 10 - \beta = 7 \\
\Rightarrow \beta = 10 - 7 \\
\Rightarrow \beta = 3 \\
 \]
Hence the value of \[\alpha = 10\] and \[\beta = 3\]
Now we know the product of two roots of a quadratic equation is given as
\[\alpha \times \beta = \dfrac{c}{a} = \dfrac{k}{1}\]
Where the value of \[\alpha = 10\] and \[\beta = 3\]
Therefore the value of k will be
\[k = \alpha \times \beta = 10 \times 3 = 30\]

Note: In the quadratic equation if \[{b^2} - 4ac > 0\] the equation will have two real roots. If it is equal, \[{b^2} - 4ac = 0\] then the equation will have only one real root and when \[{b^2} - 4ac < 0\] then the root is in complex form.