Question

Can anyone help with Discriminants in math??
(“number and types of solutions” is what the rest says)

Answer 1

#1

• 3x²-3x+2

D:-

• b²-4ac
• (-3)²-4(3)(2)
• 9-24
• -15

D<0 so unequal and un real roots

#2

• b²-4ac
• (-10)²-4(1)(1)
• 100-4
• 96

D>0 so unequal and real roots

#3

• (-4)²-4(4)(1)
• 16-16
• 0

Equal and real roots

Answer 2

Answer:

Discriminant

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

$\textsf{when }\:b^2-4ac > 0 \implies \textsf{two real solutions}$

$\textsf{when }\:b^2-4ac=0 \implies \textsf{one real solution}$

$\textsf{when }\:b^2-4ac < 0 \implies \textsf{no real solutions}$

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Question 5

Given function:  $f(x)=3x^2-3x+2$

$\implies a=3, \quad b=-3, \quad c=2$

Inputting these values into the discriminant:

$\implies \textsf{discriminant}= (-3)^2-4(3)(2)=-15$

As -15 < 0 there are no real solutions

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Question 6

Given function:  $f(x)=x^2-10x+1$

$\implies a=1, \quad b=-10, \quad c=1$

Inputting these values into the discriminant:

$\implies \textsf{discriminant}= (-10)^2-4(1)(1)=96$

As 96 > 0 there are two real solutions

at $x=5 \pm 2\sqrt{6}$

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Question 7

Given function:  $f(x)=x^2-4x+4$

$\implies a=1, \quad b=-4, \quad c=4$

Inputting these values into the discriminant:

$\implies \textsf{discriminant}= (-4)^2-4(1)(4)=0$

As 0 = 0 there is one real solution

at $x=2$