Draw the graph of the following quadratic equation and
State their nature of solutions.
x2 + x – 12 = 0
Answers
EXPLANATION.
Graph of the quadratic equation.
⇒ x² + x - 12 = 0.
As we know that,
Factorizes the equation into middle term splits, we get.
⇒ x² + 4x - 3x - 12.
⇒ x(x + 4) - 3(x + 4).
⇒ (x - 3)(x + 4).
Zeroes of the equation.
⇒ (x - 3)(x + 4) = 0.
⇒ x = 3 and x = - 4.
Put the value of x = 3 in the equation, we get.
⇒ y = x² + x - 12.
⇒ y = (3)² + (3) - 12.
⇒ y = 9 + 3 - 12.
⇒ y = 12 - 12.
⇒ y = 0.
Their Co-ordinates = (3,0).
Put the value of x = - 4 in the equation, we get.
⇒ y = x² + x - 12.
⇒ y = (-4)² + (-4) - 12.
⇒ y = 16 - 4 - 12.
⇒ y = 16 - 16.
⇒ y = 0.
Their Co-ordinates = (-4,0).
Put the value of x = 0 in the equation, we get.
⇒ y = x² + x - 12.
⇒ y = (0)² + (0) - 12.
⇒ y = - 12.
Their Co-ordinates = (0,-12).
Also we find the nature of quadratic equation.
⇒ x² + x - 12 = 0.
As we know that,
⇒ D = Discriminant Or b² - 4ac.
⇒ D = (1)² - 4(1)(-12).
⇒ D = 1 + 48.
⇒ D = 49.
Roots are rational and different, if b² - 4ac is a perfect square.
MORE INFORMATION.
Nature of the roots of the quadratic expression.
(1) = Real and unequal, if b² - 4ac > 0.
(2) = Rational and different, if b² - 4ac is a perfect square.
(3) = Real and equal, if b² - 4ac = 0.
(4) = If D < 0 Roots are imaginary and unequal Or complex conjugate.
Given :-
x² + x - 12 = 0
To Find :-
Nature
Solution :-
We know that
D = b² - 4ac
Here
b = 1
a = 1
c = -12
D = 1² - 4 × 1 × -12
D = 1 - 4 × -12
D = 1 - (-48)
D = 1 + 48
D = 49
Since
We may observe that 49 is the square of 7. Therefore, the roots are rational i.e can be represented as a fraction.
