Question

using the discriminant, state the number and type of roots.

1. ( 3x^2 - 7 = 5x )
2. ( 6x^2 = x - 1 )

Explanation:

For Problem 17: $3x^2 -7=5x$

Step1: Rewrite in standard form

Rearrange to $ax^2+bx+c=0$:
$3x^2 -5x -7=0$

Step2: Identify $a,b,c$

$a=3$, $b=-5$, $c=-7$

Step3: Calculate discriminant

Use $\Delta=b^2-4ac$:
$\Delta=(-5)^2 -4(3)(-7)=25 + 84=109$

Step4: Analyze discriminant

$\Delta=109>0$, not a perfect square.

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For Problem 18: $6x^2=x-1$

Step1: Rewrite in standard form

Rearrange to $ax^2+bx+c=0$:
$6x^2 -x +1=0$

Step2: Identify $a,b,c$

$a=6$, $b=-1$, $c=1$

Step3: Calculate discriminant

Use $\Delta=b^2-4ac$:
$\Delta=(-1)^2 -4(6)(1)=1 -24=-23$

Step4: Analyze discriminant

$\Delta=-23<0$

Answer:

1. 2 distinct irrational real roots
2. 2 distinct complex (non-real) roots