Let p and q be two positive numbers such that | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given $p + q = 2$ and $p^{4} + q^{4} = 272$. We know that $p^{2} + q^{2} = (p + q)^{2} - 2pq = 2^{2} - 2pq = 4 - 2pq$. Also,$p^{4} + q^{4} = (p^{2

Vedclass · Accepted Answer

$x^{2} - 2x + 16 = 0$

Vedclass · Answer

(D) Given $p + q = 2$ and $p^{4} + q^{4} = 272$. We know that $p^{2} + q^{2} = (p + q)^{2} - 2pq = 2^{2} - 2pq = 4 - 2pq$. Also,$p^{4} + q^{4} = (p^{2} + q^{2})^{2} - 2p^{2}q^{2} = 272$. Substituting the expression for $p^{2} + q^{2}$: $(4 - 2pq)^{2} - 2(pq)^{2} = 272$. $16 - 16pq + 4(pq)^{2} - 2(pq)^{2} = 272$. $2(pq)^{2} - 16pq - 256 = 0$. $(pq)^{2} - 8pq - 128 = 0$. Let $t = pq$. Then $t^{2} - 8t - 128 = 0$. Using the quadratic formula: $t = \frac{8 \pm \sqrt{64 - 4(1)(-128)}}{2} = \frac{8 \pm \sqrt{64 + 512}}{2} = \frac{8 \pm \sqrt{576}}{2} = \frac{8 \pm 24}{2}$. $t = 16$ or $t = -8$. Since $p$ and $q$ are positive,$pq$ must be positive,so $pq = 16$. The quadratic equation with roots $p$ and $q$ is $x^{2} - (p+q)x + pq = 0$. Substituting the values,we get $x^{2} - 2x + 16 = 0$.

Let $p$ and $q$ be two positive numbers such that $p + q = 2$ and $p^{4} + q^{4} = 272$. Then $p$ and $q$ are roots of the equation:

Similar Questions

For what value of $a$ is one root of the quadratic equation $(a^2 - 5a + 3) x^2 + (3a - 1) x + 2 = 0$ twice the other (in $/3$)?

The value of $a$ $(a \ge 3)$ for which the sum of the cubes of the roots of $x^2 - (a - 2)x + (a - 3) = 0$ assumes the least value is:

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 5x + 16 = 0$,and if $\alpha^2 + \beta^2$ and $\alpha\beta/2$ are the roots of the equation $x^2 + px + q = 0$,then:

For the equation $2x^2 + 2(a + b)x + a^2 + b^2 = 0$,if $\alpha$ and $\beta$ are the roots,then the equation whose roots are $(\alpha + \beta)^2$ and $(\alpha - \beta)^2$ is:

If $\alpha+\beta=-2$ and $\alpha^3+\beta^3=-56$,then the quadratic equation whose roots are $\alpha$ and $\beta$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module