Question

complete the following activity to determine the nature of the root of the quadratic equation 2x2 - 5x+3=0.​

Answer 1

EXPLANATION.

Nature of the quadratic equation.

⇒ 2x² - 5x + 3 = 0.

As we know that,

⇒ D = Discriminant Or b² - 4ac.

Put the value in the equation, we get.

⇒ D = (-5)² - 4(2)(3).

⇒ D = 25 - 24.

⇒ D = 1.

Nature of roots are real and equal : D = 0.

                                                                                                                           

MORE INFORMATION.

Nature of the roots of the quadratic expression.

(1) = Real and different, 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.

Answer 2

${\large{\pmb{\sf{\underline{Given \; that...}}}}}$

${\green{\bigstar}}$ A quadratic equation is given as ${\red{\sf{2x^{2} - 5x + 3 = 0}}}$

${\large{\pmb{\sf{\underline{To \; find...}}}}}$

${\green{\bigstar}}$ Determine the nature of the root of the given quadratic equation.

${\large{\pmb{\sf{\underline{Solution...}}}}}$

${\green{\bigstar}}$ The nature of the root of the quadratic equation ${\red{\sf{2x^{2} - 5x + 3 = 0}}}$ is ${\red{\sf{Real}}}$ and ${\red{\sf{Equal}}}$

${\large{\pmb{\sf{\underline{Knowledge...}}}}}$

Some knowledge about Quadratic Equations -

★ Sum of zeros of any quadratic equation is given by ➝ α+β = -b/a

★ Product of zeros of any quadratic equation is given by ➝ αβ = c/a

★ Discriminant is given by b²-4ac

• Discriminant tell us about there are solution of a quadratic equation as no solution, one solution and two solutions.

★ A quadratic equation have 2 roots

★ ax² + bx + c = 0 is the general form of quadratic equation

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━

${\large{\pmb{\sf{\underline{Using \; concepts...}}}}}$

${\red{\bigstar}}$ Discriminant is given by the given:

${\small{\underline{\boxed{\sf{b^{2} - 4ac}}}}}$

${\large{\pmb{\sf{\underline{Full \; Solution...}}}}}$

${\small{\underline{\boxed{\sf{b^{2} - 4ac}}}}}$

${\sf{:\implies b^{2} - 4ac}}$

${\sf{:\implies Given \: that \: = 2x^{2} - 5x + 3 = 0}}$

${\sf{:\implies (-5)^{2} - 4(2)(3)}}$

${\sf{:\implies -5 \times -5 - 4(2)(3)}}$

${\sf{:\implies 5 \times 5 - 4(2)(3)}}$

${\sf{:\implies 25 - 4(2)(3)}}$

${\sf{:\implies 25 - 4(6)}}$

${\sf{:\implies 25 - 4 \times 6}}$

${\sf{:\implies 25 - 24}}$

${\sf{:\implies 1}}$

Henceforth, the discriminant is 1. Therefore, the nature of the root of the given quadratic equation is Equal and Real.