3.3E: Complex Zeros of Quadratics Functions (Exercises)
- Page ID
- 99728
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Write each complex expression with the complex number \(i\).
- \(\sqrt{-9}\)
- \(\sqrt{-16}\)
- \(\sqrt{-51}\)
- \(\sqrt{-24}\)
2. Perform each operation and simplify each expression to a single complex number of the form a + bi.
- \(\sqrt{-6} \sqrt{-24}\)
- \(\sqrt{-81} - \sqrt{-4}\)
- \(3\sqrt{-25} - 4\sqrt{-64}\)
3. Simplify each expression to a single complex number of the form a + bi.
- \(\dfrac{4+\sqrt{-9} }{4}\)
- \(\dfrac{2+\sqrt{-12} }{2}\)
- \(\dfrac{4+\sqrt{-20} }{2}\)
4. Find the zeros of each quadratic functions. Write complex zeros in the form a + bi.
- \(f(x)=x^{2} -4x+13\)
- \(f(x)=x^{2} -2x+5\)
- \(f(x)=3x^{2} +2x+10\)
5. Perform each operation and simplify each expression to a single complex number of the form a + bi.
- \(\left(3+2i\right)+(5-3i)\)
- \(\left(-2-4i\right)+\left(1+6i\right)\)
- \(\left(-5+3i\right)-(6-i)\)
- \(\left(2+3i\right)(4i)\)
- \(\left(-2+4i\right)\left(8\right)\)
6. Perform each operation and simplify each expression to a single complex number of the form a + bi.
- \(\left(3+4i\right)\left(3-4i\right)\)
- \(\left(2+3i\right)(4-i)\)
- \(\left(-1+2i\right)(-2+3i)\)
- \(\left(4-2i\right)(4+2i)\)
7. The quadratic function \(y=f(x) \) has a zero \(x = 4i \) and the graph contains the point (0,5).
- What is the other zero?
- Find a formula for the quadratic function \(f(x) \) with real coefficients.
8. The quadratic function \(y=g(x) \) has a zero \(x = -3i \) and the graph contains the point (2,4).
- What is the other zero?
- Find a formula for the quadratic function \(g(x) \) with real coefficients.
9. The quadratic function \(y=h(x) \) has a zero \(x = 2 + 3i \) and the graph contains the point (3,-1).
- What is the other zero?
- Find a formula for the quadratic function \(h(x) \) with real coefficients.
10. The polynomial function \(y=k(x) \) has zeros \(x = -4 -5i \) and the graph contains the point (-1,2).
- Are there any other zeros?
- Find a formula for \(k(x) \) with real coefficients. Is \(k(x) \)a quadratic function?
- Answer
-
1. a) \(4i\)
2. b) \(5i\)
4. a) \(2 \pm 3i\)
5. a) \(8 - i\)
6. a) \(25\)
7. a) \(x = -4i\) is also a zero. b) \(f(x)= \dfrac{5}{16}x^2+5 \)