Q: Given the quadratic equation $2x^2 - 4x + 1 = 0$, what is the value of the discriminant?

$D = 4 - 4 \times 2 \times 1 = -4$

Question 1

Which of the following is the definition of a linear equation in one variable?

Accepted Answer

$ax + b = 0$ where $a$ and $b$ are real numbers

Question 2

Solve the linear equation $3x - 9 = -24$.

Accepted Answer

$x = -11$

Question 3

Which of the following is the standard form of a quadratic equation?

Accepted Answer

$ax^2 + bx + c = 0$

Question 4

What is the solution to the quadratic equation $x^2 - 5x = 0$ by factoring?

Accepted Answer

$x = 0$ or $x = 5$

Question 5

Given the quadratic equation $2x^2 - 4x + 1 = 0$, what is the value of the discriminant?

Accepted Answer

$D = 4 - 4 	imes 2 	imes 1 = -4$

Question 6

If the discriminant of a quadratic equation is negative, what can you conclude about its solutions?

Accepted Answer

There are no real solutions; the solutions are complex conjugates.

Question 7

Solve the radical equation $\sqrt{x} = 3$.

Accepted Answer

$x = 9$

Question 8

What is the principal square root of $-49$ in the complex number system?

Accepted Answer

$7i$

Question 9

Simplify the expression $(4 + 7i) + (2 + 7i)$.

Accepted Answer

$6 + 14i$

Question 10

What is the product of $(6 - 2i)(3 + i)$?

Accepted Answer

$18 - 6i + 3i - 2i^2 = 18 - 3i + 2$ (since $i^2 = -1$), so $= 20 - 3i$

Question 11

If $i$ is the imaginary unit, what is the value of $i^{23}$?

Accepted Answer

$i$

Question 12

Which of the following statements about complex conjugates is true?

Accepted Answer

The product of a complex number and its conjugate is always a real number.

Question 13

Solve the equation $x^2 + 8 = 0$ in the complex number system.

Accepted Answer

$x = \pm 2\sqrt{2}i$

Question 14

A total of $15,000 is to be divided between Jon and Wendy, with Jon to receive $2,500 less than Wendy. How much will Jon receive?

Accepted Answer

$7,250$

Question 15

Which of the following is a necessary step when solving a radical equation such as $\sqrt{x + 5} = 3$?

Accepted Answer

Isolate the radical and then square both sides.

Question 16

What is the solution to the equation $x^2 - 4x + 4 = 0$?

Accepted Answer

$x = 2$ (double root)

Question 17

If $x = 3 + 5i$ and $w = -2 - 4i$, what is $x + w$ in standard form?

Accepted Answer

$1 + i$

Question 18

Which of the following is the correct quadratic formula for solving $ax^2 + bx + c = 0$?

Accepted Answer

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Question 19

A square piece of cardboard has a side of 3 cm. If squares of side $x$ are cut from each corner and the edges are turned up to form a box with a volume of $147$ cm$^3$, what equation models the situation?

Accepted Answer

$(3 - 2x)^2 x = 147$

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