Is the following statement True or False? Jus | vedclass

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

Question

(A) The statement is True.Let the zeroes of the quadratic polynomial $p(x) = ax^2 + bx + c$ be $\alpha$ and $\beta$,where $\alpha < 0$ and $\beta < 0$

Vedclass · Accepted Answer

True

Vedclass · Answer

(A) The statement is True.Let the zeroes of the quadratic polynomial $p(x) = ax^2 + bx + c$ be $\alpha$ and $\beta$,where $\alpha From the relationship between zeroes and coefficients:$1$. The sum of the zeroes is $\alpha + \beta = -\frac{b}{a}$. Since both $\alpha$ and $\beta$ are negative,their sum $(\alpha + \beta)$ must be negative. Therefore,$-\frac{b}{a}  0$. This means $a$ and $b$ must have the same sign.$2$. The product of the zeroes is $\alpha \cdot \beta = \frac{c}{a}$. Since the product of two negative numbers is positive,$\frac{c}{a} > 0$. This means $a$ and $c$ must have the same sign.Since $a$ and $b$ have the same sign,and $a$ and $c$ have the same sign,it follows that $a, b,$ and $c$ all have the same sign.

Part $I$	Part $II$
$1.$ The graph of $p(x)=x^{2}-5x+6$	$a.$ Intersects the $X$-axis at one point.
$2.$ The graph of $p(x)=x^{2}-10x+25$	$b.$ Intersects the $X$-axis at zero points.
$3.$ The graph of $p(x)=x^{3}-4x$	$c.$ Intersects the $X$-axis at two points.
$4.$ The graph of $p(x)=x^{2}+4x+5$	$d.$ Intersects the $X$-axis at three points.

Is the following statement True or False? Justify your answer. If the zeroes of a quadratic polynomial $ax^2 + bx + c$ are both negative,then $a, b,$ and $c$ all have the same sign.

Similar Questions

The zeros of the cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$ where $a \neq 0$ and $a, b, c, d \in R$ are $\alpha, \beta,$ and $\gamma$. Then $\alpha \beta \gamma = \dots$

Divide $x^{2}+8x+12$ by $x+2$.

Given that $\sqrt{2}$ is a zero of the cubic polynomial $6x^{3}+\sqrt{2}x^{2}-10x-4\sqrt{2}$,find its other two zeroes.

State the degree of the following polynomial: $p(x) = x^{15} - (x^5)^6 + x^3 - 7$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module