ACT Prep

Front

What is a vector?

Back

A vector is a quantity that has both magnitude and direction, represented in coordinate form as \( \mathbf{u} = \langle x, y \rangle \).

Front

How do you multiply a vector by a scalar?

Back

To multiply a vector \( \mathbf{u} = \langle x, y \rangle \) by a scalar \( k \), multiply each component by the scalar: \( k\mathbf{u} = \langle kx, ky \rangle \).

Front

What is the result of \( (4 - 7i) - (-3 + 6i) \)?

Back

The result is \( 7 - 13i \).

Front

How do you solve the equation \( 3x^2 - 6 = -9 \)?

Back

First, simplify to \( 3x^2 = -3 \), then divide by 3 to get \( x^2 = -1 \), leading to \( x = \pm i \).

Front

What are complex numbers?

Back

Complex numbers are numbers that have a real part and an imaginary part, expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

Front

What is the conjugate of a complex number \( a + bi \)?

Back

The conjugate of a complex number \( a + bi \) is \( a - bi \).

Front

How do you simplify the expression \( -7i \)?

Back

The expression is already in its simplest form, representing a purely imaginary number.

Front

What is the quadratic formula?

Back

The quadratic formula is used to find the roots of a quadratic equation \( ax^2 + bx + c = 0 \) and is given by \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \).

Front

What does it mean for a number to be imaginary?

Back

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit \( i \), where \( i = \sqrt{-1} \).

Front

How do you find the roots of a polynomial equation?

Back

To find the roots of a polynomial equation, set the equation to zero and solve for the variable using factoring, the quadratic formula, or numerical methods.