Question

if d<0 the quadratic equations has two different real roots​

Answer 1

Answer :

False

Step-by-step explanation :

General form of quadratic equation : ax² + bx + c = 0

  $\boxed{\sf x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}}$

 $\boxed{\sf x=\dfrac{-b \pm \sqrt{D}}{2a}} \: \sf [since \ D=b^2-4ac]$

Nature of roots is determined by the value of discriminant (D)

• The number, D = b² - 4ac is called discriminant

1. If D > 0 ; the quadratic equation has two different real roots
2. If D = 0 ; the quadratic equation has two equal real roots
3. If D < 0 ; the quadratic equation has no real roots i.e., complex roots

Let's look at examples for each case.

1) x² + 4x - 5 = 0

By comparing, a = 1 , b = 4 , c = -5

 D = b² - 4ac

 D = 4² - 4(1)(-5)

 D = 16 + 20

 D = 36 > 0

Now, finding the roots

 $\sf x=\dfrac{-4\pm \sqrt{36} }{2} \\\\ \sf x=\dfrac{-4 \pm 6}{2} \\\\ \sf x=-2 \pm 3 \\\\ \sf x=-2+3 \: ; \: x=-2-3 \\ \sf x=1 \: ; \: -5$

1 and -5 are real roots and not equal

∴ The given equation has two distinct real roots.

2) x² + 4x + 4 = 0

a = 1 , b = 4 , c = 4

D = b² - 4ac

D = 4² - 4(1)(4)

D = 16 - 16

D = 0

Now, finding the roots

$\sf x=\dfrac{-4\pm \sqrt{0} }{2} \\\\ \sf x=\dfrac{-4 \pm 0}{2} \\\\ \sf x=\dfrac{-4}{2} \\\\ \sf x=-2$

∴ The given equation has two equal real roots.

3) x² + 4x + 5 = 0

a = 1 , b = 4 , c = 5

D = b² - 4ac

D = 4² - 4(1)(5)

D = 16 - 20

D = -4 < 0

Finding the roots,

$\sf x=\dfrac{-4\pm \sqrt{-4} }{2} \\\\ \sf x=\dfrac{-4 \pm \sqrt{-4}}{2} \\\\ \sf x=-2 \pm \dfrac{\sqrt{-4}}{2} \\\\ \sf x=-2+\dfrac{2i}{2} \: ; \: x=-2-\dfrac{2i}{2} \\\\ \sf x=-2+i \: ; \: x=-2-i$

∴ The given equation has two complex roots.

Answer 2

$\huge\sf\underline\red{Explanation}$

The given statement is false Because If d<0 Then roots are complex and conjugate to each other

NATURE OF ROOTS:-

If D>0 Roots are real and distinct

If D<0 Roots are Complex and conjugate to each other

If D =0 Roots are real and equal

______________________________

So the given statemnet is false We can find out nature of roots with help of discriminnat only

Hope u understood