Question

If 3 is a root of the equation x^2-kx+42=0 find the value of k and root of the equation

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: 3 \: is \: the \: root \: of \: the \: equation \: {x}^{2} - kx + 42 = 0$

Let assume that

$\rm \: \alpha , \beta \: be \: the \: root \: of \: the \: equation \: {x}^{2} - kx + 42 = 0$

So,

$\rm \: \alpha = 3$

We know,

$\boxed{\red{\sf Product\ of\ the\ roots=\frac{Constant}{coefficient\ of\ x^{2}}}}$

So,

$\rm \: \alpha \times \beta = \dfrac{42}{1}$

$\rm \: 3 \times \beta = 42$

$\rm\implies \: \beta = 14$

Now, we further know that

$\boxed{\red{\sf Sum\ of\ the\ roots=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

So,

$\rm \: \alpha + \beta = - \: \dfrac{( - k)}{1}$

$\rm \: 3 + 14 = k$

$\rm\implies \:k \: = \: 17$

So,

$\red{\begin{gathered}\begin{gathered}\bf\: \rm\implies \:\begin{cases} &\sf{ \beta \: = \: 14} \\ \\ &\sf{k \: = \: 17} \end{cases}\end{gathered}\end{gathered}}$

$\rule{190pt}{2pt}$

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac