Unit 4 Review — Student Classwork

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Algebra 2 Unit 4: Review

You may not use a calculator.

1. Select all expressions that are equivalent to \(4^{3/2}\).

A. \(\left(\sqrt{(4)}\right)^3\)
B. \(\sqrt{(4^3)}\)
C. \(4^3 \cdot 4\)
D. \(8\)
E. \(6\)

2. How many real solutions does \(x^2 + 2x + 5 = 0\) have?

A. 0
B. 1
C. 2

3. Select all the solutions, real or complex, to \((x - 1)^2 = -4\).

A. \(x = 3\)
B. \(x = -1\)
C. \(x = 1 + 2i\)
D. \(x = 1 - 2i\)
E. \(x = 1 + 4i\)

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4. Let \(p = 3 + 2i\) and \(q = 1 - 5i\). Write each expression in the form \(a + bi\).

a. \(p + q\)

b. \(p - q\)

c. \(pq\)

5a. Show how to solve the equation \(\sqrt{(2x + 3)} - 5 = -2\).

b. Explain why \(\sqrt{(2x + 3)} + 5 = -2\) has no real solution.

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6a. Here is a graph of \(g(x) = \sqrt[3]{(x)}\).

Use the graph to explain why there is only one real solution to every equation of the form \(\sqrt[3]{(x)} = a\) in which \(a\) is a real number.

b. Use the meaning of cube roots to show how to find an exact solution to the equation \(\sqrt[3]{(x + 1)} = 2\) without using a graph.

c. Use the meaning of cube roots to show how to find an exact solution to the equation \(\sqrt[3]{(x)} - 1 = 2\) without using a graph.

7. Noah and Lin are each trying to solve the equation \(x^2 - 4x + 5 = 0\). They know that the solutions to \(x^2 = -1\) are \(i\) and \(-i\) but are not sure how to use this to solve their equation.

a. Here is Noah's work:

\(x^2 - 4x + 5 = 0\)
\(x^2 - 4x = -5\)
\(x^2 - 4x + 4 = -5 + 4\)
\((x - 2)^2 = -1\)

Show how Noah can finish his work using complex numbers.

b. Lin decides to use the quadratic formula. Here is her work:

\(\displaystyle x = \frac{-b \pm \sqrt{(b^2 - 4ac)}}{2a}\)
\(\displaystyle x = \frac{-(-4) \pm \sqrt{((-4)^2 - 4(1)(5))}}{2(1)}\)
\(\displaystyle x = \frac{4 \pm \sqrt{(16 - 20)}}{2}\)

Lin knows \(16 - 20\) is a negative number and isn't sure what to do next. Show how Lin can write her solution using \(i\).

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Unit 4 Reference Sheet

Key terms, definitions, and rules. Use this page during the review.

Term	Definition	Rules / Outcomes
Rational Exponent \(b^{m/n}\)	A fractional exponent. The denominator is the index of the root; the numerator is the power applied to the base.	\(b^{m/n} = \left(\sqrt[n]{(b)}\right)^m = \sqrt[n]{(b^m)}\) Order: root and power can be applied in either order.
Discriminant \(b^2 - 4ac\)	The expression under the radical in the quadratic formula. Determines the number and type of solutions to \(ax^2+bx+c=0\).	If \(b^2-4ac > 0\): 2 real solutions If \(b^2-4ac = 0\): 1 real solution If \(b^2-4ac < 0\): 0 real solutions (2 complex)
Imaginary Unit \(i\)	Defined so that \(i^2 = -1\). It is the square root of \(-1\). It is not a real number.	\(\sqrt{(-a)} = \sqrt{(a)} \cdot i\) for \(a > 0\) Replace every \(i^2\) in a product with \(-1\).
Complex Number \(a + bi\)	Standard form with real part \(a\) and imaginary part \(bi\), where \(a\) and \(b\) are real numbers.	Add/Subtract: combine real parts, then imaginary parts Multiply: FOIL, then replace \(i^2\) with \(-1\)
Square Root Equation \(\sqrt{(f(x))} = k\)	An equation with a variable inside a square root. Isolate the radical, then square both sides.	If \(k \geq 0\): possible solution — check for extraneous If \(k < 0\): no real solution (\(\sqrt{\phantom{x}}\) is never negative)
Cube Root Equation \(\sqrt[3]{(f(x))} = k\)	An equation with a variable inside a cube root. Isolate the radical, then cube both sides.	Always exactly 1 real solution for any value of \(k\) No extraneous solutions.
Completing the Square \((x+p)^2 = q\)	Rewriting a quadratic so one side is a perfect square trinomial. Then take square roots of both sides.	If \(q > 0\): 2 real solutions \(x = -p \pm \sqrt{(q)}\) If \(q = 0\): 1 real solution \(x = -p\) If \(q < 0\): 2 complex solutions \(x = -p \pm i\sqrt{(-q)}\)
Quadratic Formula \(\dfrac{-b \pm \sqrt{(b^2-4ac)}}{2a}\)	Gives all solutions to \(ax^2+bx+c=0\) directly from its coefficients. Works for every quadratic.	Sign of \(b^2-4ac\) determines solution type (see Discriminant row) When \(b^2-4ac < 0\): \(\sqrt{(b^2-4ac)} = i\sqrt{(\|b^2-4ac\|)}\)