Question

2) Solve the following quachatic polyro
mials gaphically x2-x-6
​

Answer 1

EXPLANATION.

Quadratic equation,

⇒ F(x) = x² - x - 6.

As we know that,

Factorizes the equation into middle term splits, we get.

⇒ y = x² - x - 6.

⇒ x² - x - 6 = 0.

⇒ x² - 3x + 2x - 6 = 0.

⇒ x(x - 3) + 2(x - 3) = 0.

⇒ (x + 2)(x - 3) = 0.

⇒ x = -2 and x = 3.

Put the value of x = -2 in equation, we get.

⇒ y = (-2)² - (-2) - 6.

⇒ y = 4 + 4 - 6.

⇒ y = 8 - 6.

⇒ y = 2.

Their Co-ordinates = (0,2).

Put the value of x = 3 in equation, we get.

⇒ y = (3)² - (3) - 6.

⇒ y = 9 - 3 - 6.

⇒ y = 9 - 9.

⇒ y = 0.

Their Co-ordinates = (3,0).

Put the value of x = 0 in equation, we get.

⇒ y = (0)² - (0) - 6.

⇒ y = -6.

Their Co-ordinates = (0,-6).

Answer 2

Solve graphically :-

$\tt \ \: :  ⟼ {x}^{2} - x - 6$

─━─━─━─━─━─━─━─━─━─━─━─━─

$\large\underline\purple{\bold{Solution :- }}$

$\tt \ \: :  ⟼ \: Let \: y \: = {x}^{2} - x - 6$

$\large\underline{\bold{❥︎Step :- 1 }}$

☆ Point of intersection with x - axis.

☆ We know, on x - axis, y = 0

$\tt\implies \: {x}^{2} - x - 6 = 0$

$\tt \ \: :  ⟼ {x}^{2} - 3x + 2x - 6 = 0$

$\tt \ \: :  ⟼ x(x - 3) + 2(x - 3) = 0$

$\tt \ \: :  ⟼ (x - 3)(x + 2) = 0$

$\tt \ \: :  ⟼ x - 3 = 0 \: or \: x + 2 = 0$

$\tt\implies \:x = 3 \: or \: x \: = - 2$

Hᴇɴᴄᴇ,

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

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

$\large\underline{\bold{❥︎Step :- 2 }}$

☆ To find the vertices of given quadratic polynomial

$\tt \ \: :  ⟼ y = {x}^{2} - x - 6$

$\tt \ \: :  ⟼ differentiate \: w.r.t. \: x \: we \: get$

$\tt \ \: :  ⟼ \dfrac{dy}{dx} = 2x - 1$

$\tt \ \: :  ⟼ for \: turning \: point \: \: \: \: \dfrac{dy}{dx} = 0$

$\tt\implies \:2x - 1 = 0$

$\tt\implies \:x = \dfrac{1}{2}$

$\tt \ \: :  ⟼ \therefore \: y \: = {(\dfrac{1}{2}) }^{2} - \dfrac{1}{2} - 6$

$\tt\implies \:y = \dfrac{1}{4} - \dfrac{1}{2} - 6$

$\tt\implies \:y = \dfrac{1 - 2 - 24}{4}$

$\tt\implies \:y = - \dfrac{25}{4}$

$\tt \ \: :  Hence, \: vertex \: is \: (\dfrac{1}{2} , - \dfrac{25}{4} )$

➢ Now draw a graph using the points

➢ See the attachment graph.

➢ Solution is x = 3 or x = - 2