Question

solve and graph

solve and graph ​

Answer 1

Answer:

4x²-9x+2<0

on factorising we get :-

(4x-1)(x-2)<0

on solving we get :-

1/4<x<2

Step-by-step explanation:

.______.

<______-1_______0___1/4____1______2__>

(.____.) means number line is from 1/4 to 1

Answer 2

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

Given quadratic inequality is

$\rm \: {4x}^{2} - 9x + 2 < 0$

Let we first plot the graph of quadratic equation

$\rm \: {4x}^{2} - 9x + 2 = 0$

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 parabola of quadratic polynomial ax² + bx + c is given by

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

Here,

$\rm \: a = 4$

$\rm \: b = - 9$

$\rm \: c = 2$

So, on substituting the values, we get

$\rm \: \:Vertex = \bigg(\dfrac{9}{8} , \: \dfrac{4(4)(2) - {( - 9)}^{2} }{4(4)} \bigg)$

$\rm \: \:Vertex = \bigg(\dfrac{9}{8} , \: \dfrac{32 - 81 }{16} \bigg)$

$\rm \: \:Vertex = \bigg(\dfrac{9}{8} , \: \dfrac{ -49}{16} \bigg)$

Step :- 2

Point of intersection with x - axis

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

So, on substituting y = 0, in given equation we get

$\rm \: {4x}^{2} - 9x + 2 = 0$

$\rm \: {4x}^{2} - 8x - x + 2 = 0$

$\rm \: 4x(x - 2) - 1(x - 2) = 0$

$\rm \: (x - 2)(4x - 1) = 0$

$\rm\implies \:x = 2 \: \: or \: \: x = \dfrac{1}{4}$

Hence, the point of intersection with x- axis is (2, 0) and ( 0.25, 0).

Now,

Point of intersection with y - axis.

We know, on y - axis, x = 0

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

$\rm \: y = 0 - 0 + 2$

$\rm\implies \:y = 2$

Hence, the point of intersection with y- axis is (0, 2).

Hᴇɴᴄᴇ,

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

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 2 \\ \\ \sf 2 & \sf 0 \\ \\ \sf 0.25 & \sf 0\\ \\ \sf \dfrac{9}{8} & \sf - \dfrac{49}{16} \end{array}} \\ \end{gathered}$

➢ Now draw a graph using the points.

➢ See the attachment graph.

So, the solution set is 0.25 < x < 2.