Question

Find the value of the discriminant. determine the number and type of solutions of the following quadratic equation without solving.

-6x^2+7x+3=0

Answer 1

Quadratic equation -

`\: \:\:\:\: \small \underline{ \boxed{ \sf{ -6x^2 +7x+3 =0}}}\\`

Where-

• a =-6
• b=7
• c =3

To find the Discriminant of this equation is given by -

`\longrightarrow\underline\purple{\boxed{\pmb{D = b^2-4ac}}}`

On substituting the values -

`\: \:\:\:\: \longrightarrow\sf { D = \bigg(7\bigg)^2 - 4* -6* 3}\\`

`\: \:\:\:\: \longrightarrow\sf { D = 49+72}\\`

`\: \:\:\:\: \longrightarrow\pmb{ \underline{\purple{D =121}}}\\`

Since, D>0 hence, this equation has two distinct real roots.

Answer 2

2 real and irrational roots

Explanation:

given a quadratic equation in standard form

ax² + bx + c = 0 ( a ≠ )

then the nature of the roots can be determined using the discriminant

Δ = b² - 4ac

• if b² - 4ac > 0 then 2 real and irrational roots

• if b² - 4ac > 0 and a perfect square then 2 real and rational roots

• if b² - 4ac = 0 then 2 real and equal roots

• if b² - 4ac < 0 then 2 complex roots

- 6x² + 7x + 3 = 0 ← is in standard form

with a = - 6 , b = 7 , c = 3

b² - 4ac = 7² - (4 × - 6 × 3) = 49 - (- 72) = 49 + 72 = 121

since b² - 4ac > 0 then the equation has 2 real and irrational roots