Math Core

This test covers the whole unit: forms of quadratic functions, solving quadratic equations, complex numbers and their arithmetic, quadratics with complex solutions, quadratic inequalities, and linear-quadratic systems.

Question 1

Write $f(x) = x^2 - 8x + 10$ in vertex form and give the vertex $(h, k)$ .

Enter a point like (2, -3)

Question 2

A parabola has zeros $-3$ and $1$ and a y-intercept of $6$ . Its equation is $y = a(x + 3)(x - 1)$ . Find $a$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 3

Solve $2x^2 + 7x - 4 = 0$ .

Separate answers with commas, e.g. 2, -5

Question 4

Find all real solutions of $x^4 - 10x^2 + 9 = 0$ .

Separate answers with commas, e.g. 2, -5

Question 5

Which describes the solutions of $3x^2 - 2x + 4 = 0$ ?

Question 6

Write $\sqrt{-75}$ in the form $k\,i\sqrt{3}$ . What is $k$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 7

Simplify $i^{30}$ .

Question 8

Simplify $(3 - 2i)(5 + 4i)$ and write it as $a + bi$ . Enter $(a, b)$ .

Enter a point like (2, -3)

Question 9

Write $\dfrac{7 + i}{1 - i}$ in the form $a + bi$ . Enter $(a, b)$ .

Enter a point like (2, -3)

Question 10

The solutions of $x^2 + 8x + 25 = 0$ are $a \pm bi$ with $b > 0$ . Enter $(a, b)$ .

Enter a point like (2, -3)

Question 11

The equation $x^2 + bx + c = 0$ has real coefficients and one solution $2 + 5i$ . What is $c$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 12

Solve $x^2 + 4x - 12 < 0$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 13

Solve $2x^2 - x - 1 \ge 0$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 14

What is the solution set of the system $y = x^2 - 3$ and $y = 2x$ ?

Question 15

For what value of $k$ is the line $y = -2x + k$ tangent to the parabola $y = x^2 + 2$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.