Question

If the sum of squares of the zeroes of the polynomial $f\left( x \right)={{x}^{2}}-8x+k$ is 40 then find the value of k.

Answer 1

Hint: Now we know that for the equation of the form $a{{x}^{2}}+bx+c$ the sum of the roots is given by $\dfrac{-b}{a}$ and the product of the roots is given by $\dfrac{c}{a}$ . Now consider the equation obtained by sum of the roots and square the equation on both sides. Now we will substitute the values from the given equation and equation obtained by the condition of the product of roots and simplify the equation. Hence we will get the value of k.

Complete step by step solution:
Now consider the given quadratic equation ${{x}^{2}}-8x+k$ .
Let $\alpha $ and $\beta $ be the roots of the given quadratic equation.
Now we are given that the sum of the squares of the zeros of polynomial is 40
Hence we get, ${{\alpha }^{2}}+{{\beta }^{2}}=40..................\left( 1 \right)$
Now again consider the given quadratic equation ${{x}^{2}}-8x+k$
Comparing the equation with the general equation of quadratic $a{{x}^{2}}+bx+c$ we get a = 1, b = -8 and c = k.
Now we know that the sum of the roots is given by $\dfrac{-b}{a}$ and the product of the roots is given by $\dfrac{c}{a}$ .
Hence we get, $\alpha +\beta =\dfrac{-\left( -8 \right)}{1}$ and $\alpha \beta =\dfrac{k}{1}$
$\Rightarrow \alpha +\beta =8.................\left( 2 \right)$
$\Rightarrow \alpha \beta =k................\left( 3 \right)$
Now squaring the equation (2) we get,
$\Rightarrow {{\left( \alpha +\beta \right)}^{2}}={{8}^{2}}$
Now we know that ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ hence using this we get,
$\Rightarrow {{\alpha }^{2}}+{{\beta }^{2}}+2\alpha \beta =64$
Now substituting the value of $\alpha \beta $ from equation (3) and value of ${{\alpha }^{2}}+{{\beta }^{2}}$ from (1) we get,
$\Rightarrow 40+2k=64$
Now subtracting 40 on both sides of the equation we get,
$\Rightarrow 2k=24$
Now dividing the whole equation by 2 we get, k = 12.

Hence the value of k is 12.

Note: Now note that we can check the solution by substituting the value of k in the equation and finding the roots of the equation by formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence we will check if the conditions of the roots are satisfied.