Question

03. If one zero of the polynomial 5x²+13x+p is the reciprocal of other find p.​

Answer 1

EXPLANATION.

One zeroes of the polynomial 5x² + 13x + p is reciprocal to other,

Let one roots = α.

other roots is = 1/α

Products of zeroes of quadratic equation,

⇒ αβ = c/a.

⇒ α X 1/α = p/5.

⇒ 1 = p/5.

⇒ p = 5.

Value of p = 5.

                                   

MORE INFORMATION.

Graph of a quadratic equation.

We have y = f(x) = ax² + bx + c where a, b, c ∈ R , a ≠ 0.

(1) = The shape of the curve y = f(x) is a parabola.

(2) = The axis of the parabola is parallel to y-axis.

(3) = If a > 0, then the parabola open upwards.

(4) = If a < 0, then the parabola open downwards.

(5) = For D > 0, parabola cuts x-axis in two distinct points.

(6) = For D = 0, parabola touches x-axis in one point.

(7) = For D < 0, parabola does not cut x-axis.

Answer 2

Answer:-

$\small \bf \underline{Given:}$

One zero of the polynomial 5x²+13x+p is the reciprocal of other

$\small \bf \underline{To \: find:}$

We need to find the value of p

$\small \bf \underline{Solution:}$

The given polymonial is,

$\bf \: f(x) = 5 {x}^{2} + 13x + p$

The quadratic equation :

$\bf \: a {x}^{2} + bx + c = 0$

Product of roots of a quadratic equation = $\bf \frac{c}{a}$

After substituting the values, the equation stands as follows :

$\bf \: 5 {x}^{2} + 13x + p = 0$

Product of the roots = $\bf \frac{p}{5}$

Given that the roots are reciprocal to each other

So if one root is n the other root is $\bf \frac{1}{n}$

Therefore, there product will always be equal to 1

$\bf1 = \frac{p}{5}$

$\bf \implies \: p = 5$

Answer: The value of p = 5

$\small \bf \underline{Short\:cut\:method:}$

If n and $\bf \frac{1}{n}$ are the roots of the f(x) = ax^2 + bx + c

then a = c

=> p = 5

Answer: The value of p = 5

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