Question

the discriminant of a quadratic equation is 24. the roots 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 $ax^2 + 4x + 2 = 0$ are real and equal when $a$ is equal to
(1) 1 (2) 2 (3) 3 (4) 4
your answer

Explanation:

Step1: Analyze discriminant rule (Q1)

For quadratic $ax^2+bx+c=0$, discriminant $D=b^2-4ac$:

• $D>0$: real, unequal roots
• $D$ perfect square: rational roots; else irrational

Given $D=24>0$, 24 is not a perfect square.

Step2: Match to root type (Q1)

Roots are real, irrational, unequal.

Step3: Set discriminant to 0 (Q2)

For real equal roots, $D=0$.
Here $b=4$, $a=a$, $c=2$, so:
$$4^2 - 4(a)(2) = 0$$

Step4: Solve for $a$ (Q2)

$$16 - 8a = 0$$
$$8a=16$$
$$a=\frac{16}{8}=2$$

Answer:

1. (4) real, irrational, and unequal
2. (2) 2