Question

the roots of the equation $9x^2 + 3x - 4 = 0$ are:
(1) imaginary
(3) real, rational, and unequal
(2) real, rational, and equal
(4) real, irrational, and unequal
your answer:

the roots of the equation $x^2 - 10x + 25 = 0$ are:
(1) imaginary
(3) real, irrational, and unequal
(2) real and irrational
(4) real, rational, and equal

Explanation:

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First Equation: $9x^2 + 3x - 4 = 0$

Step1: Identify coefficients

For $ax^2+bx+c=0$, $a=9$, $b=3$, $c=-4$

Step2: Calculate discriminant

Discriminant formula: $\Delta = b^2-4ac$
$\Delta = 3^2 - 4(9)(-4) = 9 + 144 = 153$

Step3: Analyze discriminant

$\Delta=153>0$, not a perfect square. So roots are real, irrational, unequal.

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Second Equation: $x^2 - 10x + 25 = 0$

Step1: Identify coefficients

For $ax^2+bx+c=0$, $a=1$, $b=-10$, $c=25$

Step2: Calculate discriminant

$\Delta = b^2-4ac$
$\Delta = (-10)^2 - 4(1)(25) = 100 - 100 = 0$

Step3: Analyze discriminant

$\Delta=0$, so roots are real, rational, equal.

Answer:

1. For $9x^2 + 3x - 4 = 0$: (4) real, irrational, and unequal
2. For $x^2 - 10x + 25 = 0$: (4) real, rational, and equal