Question

Solve the discriminant

Answer 1

a

Explanation:

given a quadratic equation in standard form

ax² + bx + c = 0 ( a ≠ 0 )

then the discriminant

Δ = b² - 4ac

• if b² - 4ac > 0 then 2 real solutions

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

• if b² - 4ac < 0 then no real solutions

given

`(3)/(4)` x² - 3x = - 4 ( add 4 to both sides )

`(3)/(4)` x² - 3x + 4 = 0 ← in standard form

with a =
`(3)/(4)` b = - 3 , c = 4

then

b² - 4ac = (- 3)² - ( 4 ×
`(3)/(4)` × 4) = 9 - 12 = - 3

since b² - 4ac < 0 then equation has no real solutions

Answer 2

a. -3; no real solutions.

Explanation:

Discriminant

`\boxed{b^2-4ac }\quad\textsf{when}\:ax^2+bx+c=0`

`\textsf{When }\:b^2-4ac > 0 \implies \textsf{two real solutions}.`

`\textsf{When }\:b^2-4ac=0 \implies \textsf{one real solution}.`

`\textsf{When }\:b^2-4ac < 0 \implies \textsf{no real solutions}.`

Given equation:

`(3)/(4)x^2-3x=-4`

Add 4 to both sides of the equation so that it is in standard form:

`\implies (3)/(4)x^2-3x+4=-4+4`

`\implies (3)/(4)x^2-3x+4=0`

Therefore, the variables are:

`a=(3)/(4), \quad b=-3, \quad c=4`

Substitute these values into the discriminant formula to find the value of the discriminant:

`\begin{aligned}\implies b^2-4ac&=(-3)^2-4\left((3)/(4)\right)(4)\\&=9-(3)(4)\\&=9-12\\&=-3\\\end{aligned}`

Therefore, as -3 < 0, the discriminant is less than zero.

This means there are no real solutions.