Question

Characterize the roots of the following quadratic equations using the discriminant
x²+5x+4=0
​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{Equation\;is\;x^2+5x+4=0}$

$\underline{\textsf{To find:}}$

$\textsf{Characterization of roots of the given equation}$

$\underline{\textsf{Solution:}}$

$\mathsf{Concept:}$

$\mathsf{The\;discriminant\;of\;the\;quadratic\;equation\;ax^2+bx+c=0\;is\;\boxed{\bf\triangle=b^2-4ac}}$

$\mathsf{Consider,\;x^2+5x+4=0}$

$\mathsf{Here,\;a=1,\;b=5,\;c=4}$

$\mathsf{Now,}$

$\mathsf{\triangle=b^2-4ac}$

$\mathsf{\triangle=5^2-4{\times}1{\times}4}$

$\mathsf{\triangle=25-16}$

$\mathsf{\triangle=9,\;a\;perfect\;square}$

$\therefore\textsf{Roots are real and rational}$

$\underline{\textsf{Find more:}}$

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Answer 2

We know,

• If discriminant (of an quadratic equation) is equal to 0, then, roots of the equation is real and equal numbers.
• If discriminant (of an quadratic equation) is less than 0, then, roots of the equation is non-real numbers.
• If discriminant (of an quadratic equation) is more than 0, then, roots of the equation is two real and distinct numbers.

Given,

$\huge {x}^{2} + 5x + 4 = 0$

• It is of the form ax² + bx + c = 0

So,

• a = 1, b = 5, and c = 4

Here,

$\text{Discriminant} = \sqrt{{b}^{2} - 4ac}$

$D = \sqrt{ {(5)}^{2} + 4(1)(4) } \\D = \sqrt{25 + 16} \\ D= \sqrt{41}$

As,

• Discriminant is √41 or 6.4 (approximate value) i.e. more than zero, so the equation has two real & distinct roots.

Thus,

• x² + 5x + 4 = 0 has two real roots.