Unit 2.2 Quiz Review PreCal

Imaginary Number

A complex number written as a real number multiplied by the imaginary unit i.

Complex Number

A real number including the imaginary unit i. They can be expressed as "a+bi".

Try it! Find the vertex of the quadratic 3x^2+6x-9

Answer: (-1,-12) Explanation: Use the formula for the x-value of the quadratic, -b/2a. This gives you -6/2*3. =-6/2*3 =-6/6. X=-1 From there, plug -1 into the original equation. =3x^2+6x-9 =3(-1^2)+6(-1)-9 =3(1)-6-9 =-12 We know know y=-12 and x=-1, making our vertex (-1,-12).

Try it! Find the vertex of the quadratic 5x^2+4x-3

Answer: (-2/5, -19/5) Explanation: Use the formula for the x-value of the quadratic, -b/2a. This gives you -4/2*5 =-4/2*5 =-4/10 =-2/5 From there, plug -2/5 into the original equation =5((-2/5)^2)+4(-2/5)-3 =5(-4/25)+4(-2/5)-3 =-(20/25)-(-8/5)-3 From here, find a common denominator. For this equation, our common denominator will be 5. =-(4/5)-(8/5)-(15/5) =-19/5 We know know our X and Y values, creating the vertex (-2/5,-19/5).

Try it! (7+2i)/(3+3i)

Answer: (27-15i)/18 Explanation: We must first multiple both parts of the fraction by the Complex Conjugate =(7+2i)/(3+3i) =[(7+2i)/(3+3i)]*[(3-3i)/(3-3i)] From here, foil both top and bottom =(21-21i+6i-6i^2)/(9-9i+9i-9i^2) =(21-15i+6)/(9+9) =(27-15i)/18

Try it! (5-8i)+(-3+9i)

Answer: 2+i Explanation: When combining complex numbers, treat real and imaginary components separately. 5+-3=2, and -8i+9i=i. This gives the final solution 2+i.

Try it! (2+5i)(4+i)

Answer: 3+22i Explanation: To multiply multiple complex numbers, we must use FOIL. =(2+5i)(4+i) =8+2i+20i+5(i^2) Remember (i^2)=-1 =8+22i+5(-1) =3+22i

Try it! (-3+4i)-(-6+2i)

Answer: 3+2i Explanation: When combining complex numbers, treat real and imaginary components separately. -3--6=3, and 4i-+2i=2i. This gives the final solution 3+2i.

Try it! Find the Complex Conjugate (6+i)/(3-4i)

Answer: 3+4i Explanation. The complex conjugate is simply the complex denominator with the imaginary unit having a sign change. This means -4i becomes +4i.

Try it! Derive the Quadratic Formula ax^2+bx+c

Answer: Quadratic Formula Explanation: Start with ax^2+bx+c, and subtract c ax^2+bx=-c Now divide by a x^2+(b/a)x=-c/a Now complete the square x^2+(b/a)x+(b^2/4a^2)=(-c/a)+(b^2/4a^2) Factor Out (x+(b/2a))^2=(-c/a)+(b^2/4a^2) Square Root Both x+(b/2a)=±(square root of (-c/a)+(b^2/4a^2)) Now get a common denominator x+(b/2a)=±(square root of (-4ac/4a^2)+(b^2/4a^2)) Now Combine x+(b/2a)=±(square root of b^2-4ac)/4a^2) Simplify x+(b/2a)=±(square root of b^2-4ac)/2a Now subtract (b/2a) x=(-b±(square root of b^2-4ac))/2a And this is the final answer

Try it! Find the Roots of the parabola x^2+12x+32

Answer:X=-8,-4 Explanation Factor out x^2+12x+32 This gives (x+8)(x+4) Now set both of these equal to 0 to find the values of X. This gives the final answers of -8 and -4

Completing the Square

By altering the equation of a quadratic to move the constant value (ax^2+bx=c), the square can be completed by dividing b by 2, and squaring that value. This new value is then added to both sides. Example: x^2+6x+6=0 x^2+6x=-6 x^2+6x+9=3 (6/2=3, and 3^2=9. This value is added to both sides of the equation). You would then solve.

Roots

Roots, or "Zeros", of a parabola are the point's in which the parabola crosses the X-axis, in other words, when X=0. Factoring is typically used to find these. Parabolas may have one, two, or no roots.

Vertex of a Quadratic

The Vertex of a Quadratic function is the lowest/highest point in the parabola. Using the values from ax^2+bx+c, the formula of -b/2a can be used to find X. X can then be plugged into ax^2+bx+c to find Y, giving (X,Y) at the Vertex.

Axis of symmetry

The axis of the symmetry is the vertical line that cuts a parabola into two symmetric halves. This line shares the same X value with the parabola's vertex Remember, a complex number is expressed as "a+bi". This is important as complex numbers that have an "a" value the same as the axis' X-Value will always produce real solutions.

Complex Conjugate

Used in creating real denominators, the Complex Conjugate is the complex number that is multiplied onto a fraction when there is a non-real number in the denominator. This number is the same complex number that is present in the denominator, but the sign of the imaginary component has been changed.

Derive the Quadratic Formula using ax^2+bx+c

Using the equation ax^2+bx+c=0, the formula x=(-b±[square root of b^2-4ac])/2a is derived.

i

i is the symbol for the imaginary unit. i has properties that states (i^2)=-1. This makes (square root of -1)=i

The Quadratic Formula

x=(-b±[square root of b^2-4ac])/2a Values are derived from ax^2+bx+c