Question

draw the graph of quadrilateral polynomial x^2+5x+6 and also find their zeros to graph​

Answer 1

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

The given quadratic polynomial is

$\rm :\longmapsto\: {x}^{2} + 5x + 6$

Let assume that

$\rm :\longmapsto\: y = {x}^{2} + 5x + 6$

To plot the graph of the quadratic polynomial which is always parabola, the following steps have to be followed :-

Step :- 1 Vertex of parabola

We know, vertex of quadratic polynomial ax² + bx + c is given by

$\blue{ \boxed{\bf \:Vertex = \bigg( - \dfrac{ b}{2a} , \: \dfrac{4ac - {b}^{2} }{4a} \bigg)}}$

Here,

$\rm :\longmapsto\:a = 1$

$\rm :\longmapsto\:b = 5$

$\rm :\longmapsto\:c = 6$

So, on substituting all these values, we get

$\rm :\longmapsto\:\:Vertex = \bigg( - \dfrac{ 5}{2(1)} , \: \dfrac{4(1)(6) - {5}^{2} }{4(1)} \bigg)$

$\rm :\longmapsto\:\:Vertex = \bigg( - \dfrac{ 5}{2} , \: \dfrac{24 - 25 }{4} \bigg)$

$\rm :\longmapsto\:\:Vertex = \bigg( - \: \dfrac{ 5}{2} , \: - \: \dfrac{1 }{4} \bigg)$

Step :- 2 Point of intersection with x - axis

We know, on x - axis, y = 0.

So, on substituting the value in given curve we get

$\rm :\longmapsto\: {x}^{2} + 5x + 6 = 0$

$\rm :\longmapsto\: {x}^{2} + 3x + 2x + 6 = 0$

$\rm :\longmapsto\:x(x + 3) + 2(x + 3) = 0$

$\rm :\longmapsto\:(x + 3)(x + 2) = 0$

$\rm :\longmapsto\:x = - 3 \: \: \: or \: \: \: x = - 2$

Hence, the point of intersection with x- axis is (- 3, 0) and ( - 2, 0).

Now,

Step :- 3 Point of intersection with y - axis

We know, on y - axis, x = 0

So, on Substituting the value in given curve, we get

$\rm :\longmapsto\:y = {0}^{2} + 5(0) + 6$

$\rm :\longmapsto\:y = 6$

Hence, the point of intersection with y- axis is (0, 6).

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 6 \\ \\ \sf - 3 & \sf 0 \\ \\ \sf - 2 & \sf 0\\ \\ \sf - \dfrac{5}{2} & \sf - \dfrac{1}{4} \end{array}} \\ \end{gathered}$

➢ Now draw a graph using the points.

➢ See the attachment graph.

From graph, we conclude that,

Zeroes of given quadratic polynomial are - 3 and - 2.