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Common Core: High School - Number and Quantity : High School: Number and Quantity

Study concepts, example questions & explanations for Common Core: High School - Number and Quantity

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All Common Core: High School - Number and Quantity Resources

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Example Questions

Example Question #81 : High School: Number And Quantity

Find the product of the complex number and its conjugate.

20i

Possible Answers:

400

-20i

Correct answer:

400

Explanation:

$20i\cdot -20i$

$-400i^{2}$

$-400\cdot -1$

400

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Example Question #82 : High School: Number And Quantity

Simplify.

$\frac{2+i}{4-i}$

Possible Answers:

$\frac{1}{2-1}$

$\frac{7+6i}{15}$

$\frac{7+6i}{4-i}$

Correct answer:

$\frac{7+6i}{15}$

Explanation:

$\frac{2+i}{4-i}\cdot \frac{4+i}{4+i}$

$=\frac{\left ( 2+i \right )\left ( 4+i \right )}{\left ( 4-i \right )\left ( 4+i \right )}$

$=\frac{8+2i+4i+i^{2}}{16+4i-4i-i^{2}}$

$=\frac{8+6i-1}{16-1}$

$=\frac{7+6i}{15}$

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Example Question #83 : High School: Number And Quantity

Simplify.

$\frac{3+2i}{9-6i}$

Possible Answers:

$\frac{1}{3}$

3+3i

$\frac{5+12i}{39}$

-6+8i

Correct answer:

$\frac{5+12i}{39}$

Explanation:

$\frac{3+2i}{9-6i}\cdot \frac{9+6i}{9+6i}$

$=\frac{\left ( 3+2i \right )\left ( 9+6i \right )}{\left ( 9-6i \right )\left ( 9+6i \right )}$

$=\frac{27+18i+18i+12i^{2}}{81+54i-54i-36i^{2}}$

$=\frac{27+36i-12}{81+36}$

$=\frac{15+36i}{117}$

$=\frac{3\left ( 5+12i \right )}{3\left ( 39 \right )}$

$=\frac{5+12i}{39}$

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Example Question #84 : High School: Number And Quantity

Simplify.

$\frac{6-4i}{-3i}$

Possible Answers:

$\frac{4+6i}{3}$

6-i

-2i

6-7i

Correct answer:

$\frac{4+6i}{3}$

Explanation:

$\frac{6-4i}{-3i}\cdot \frac{3i}{3i}$

$\frac{18i-12i^{2}}{-9i^{2}}$

$\frac{18i+12}{9}$

$\frac{12+18i}{9}$

$\frac{3\left ( 4+6i \right )}{3\left ( 3 \right )}$

$\frac{4+6i}{3}$

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Example Question #1 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Solve for .

4x^2+4x+2

Possible Answers:

$x=\frac{1}{2}\pm\frac{i}{4}$

$x=\frac{-1}{4}\pm\frac{i}{4}$

$x=\frac{-1}{2}\pm\frac{i}{2}$

$x=\frac{1}{4}\pm\frac{i}{4}$

$x=\frac{1}{2}\pm\frac{i}{2}$

Correct answer:

$x=\frac{-1}{2}\pm\frac{i}{2}$

Explanation:

This question is testing one's ability to solve quadratic equations using the quadratic formula. Furthermore, it builds one's ability to recognize and understand properties of parabolas, dealing with quadratics, and identifying roots.

For the purpose of Common Core Standards, "solve quadratic equations with real coefficients that have complex solutions", falls within the Cluster C of "use complex numbers in polynomial identities and equations" (CCSS.MATH.CONTENT.HSF.CN.C).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: State the general quadratic formula.

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

where the quadratic function is in the form,

ax^2+bx+c .

Step 2: Identify the coefficients.

4x^2+4x+2

$\\a=4 \\b=4 \\c=2$

Step 3: Substitute coefficients into the quadratic formula.

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{-4\pm\sqrt{4^2-4(4)(2)}}{2(4)}$

$x=\frac{-4\pm\sqrt{16-32}}{8}$

$x=\frac{-4\pm\sqrt{-16}}{8}$

Recall that a negative sign under the radical represents an imaginary number.

$x=\frac{-4\pm4i}{8}$

$x=\frac{-4}{8}\pm\frac{4i}{8}$

$x=\frac{-1}{2}\pm\frac{i}{2}$

This solution shows that the roots of this particular quadratic are imaginary.

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Example Question #1 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Solve.

$3x^{2}-2x+7$

Possible Answers:

$x=\frac{1\pm 2i\sqrt{5}}{3}$

$x=3\textup{ and }x=-2$

$x=\pm 2$

$x=1\pm 5\sqrt{2}$

Correct answer:

$x=\frac{1\pm 2i\sqrt{5}}{3}$

Explanation:

a=3

b=-2

c=7

$\frac{2\pm \sqrt{\left ( -2 \right )^{2}-4\cdot 3\cdot 7}}{2\cdot 3}$

$\frac{2\pm \sqrt{4-84}}{6}$

$\frac{2\pm \sqrt{-80}}{6}$

$\frac{2\pm \sqrt{16\cdot 5\cdot -1}}{6}$

$\frac{2\pm 4i\sqrt{5}}{6}$

$\frac{1\pm 2i\sqrt{5}}{3}$

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Example Question #3 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Solve.

$4x^{2}+2x+3$

Possible Answers:

$x=\pm 2i$

$x=-2\pm \sqrt{11}$

$x=\frac{-2\pm i\sqrt{11}}{4}$

$x=4\textup{ and }x=-1$

Correct answer:

$x=\frac{-2\pm i\sqrt{11}}{4}$

Explanation:

a=4

b=2

c=3

$x=\frac{-4\pm \sqrt{\left ( 2 \right )^{2}-4\cdot 4\cdot 3}}{2\cdot 4}$

$x=\frac{-4\pm \sqrt{4-48}}{8}$

$x=\frac{-4\pm \sqrt{-44}}{8}$

$x=\frac{-4\pm \sqrt{4\cdot 11\cdot -1}}{8}$

$x=\frac{-4\pm 2i\sqrt{11}}{8}$

$x=\frac{-2\pm i\sqrt{11}}{4}$

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Example Question #85 : High School: Number And Quantity

Solve.

$x^{2}-2x+5$

Possible Answers:

$x=1\pm 2i$

$x=\pm 2i$

$x=1\textup{ and }x=-2$

$x=1\pm \sqrt{16}$

Correct answer:

$x=1\pm 2i$

Explanation:

$x=\frac{-(-2)\pm \sqrt{(-2)^{2}-4\cdot 1\cdot 5}}{2\cdot 1}$

$x=\frac{2\pm \sqrt{4-20}}{2}$

$x=\frac{2\pm \sqrt{-16}}{2}$

$x=\frac{2\pm 4i}{2}$

$x=1\pm 2i$

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Example Question #5 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Solve.

$a^{2}-6a+13$

Possible Answers:

$a=3\pm 2i$

$a=\pm 3$

$a=3\textup{ and }a=-1$

$a=\pm 4i$

Correct answer:

$a=3\pm 2i$

Explanation:

$a=\frac{-(-6)\pm \sqrt{\left ( -6 \right )^{2}-4\cdot 1\cdot 13}}{2\cdot 1}$

$a=\frac{6\pm \sqrt{36-52}}{2}$

$a=\frac{6\pm \sqrt{-16}}{2}$

$a=\frac{6\pm \sqrt{16\cdot -1}}{2}$

$a=\frac{6\pm 4i}{2}$

$a=3\pm 2i$

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Example Question #2 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Solve.

$3b^{2}-4b+5$

Possible Answers:

$b=\frac{2\pm i\sqrt{11}}{3}$

$b=\pm 3$

$b=2\pm i\sqrt{11}$

$b=\pm 4i$

Correct answer:

$b=\frac{2\pm i\sqrt{11}}{3}$

Explanation:

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Common Core: High School - Number and Quantity : High School: Number and Quantity

Study concepts, example questions & explanations for Common Core: High School - Number and Quantity

All Common Core: High School - Number and Quantity Resources

Example Questions

Example Question #81 : High School: Number And Quantity

Example Question #82 : High School: Number And Quantity

Example Question #83 : High School: Number And Quantity

Example Question #84 : High School: Number And Quantity

Example Question #1 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Example Question #1 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Example Question #3 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Example Question #85 : High School: Number And Quantity

Example Question #5 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Example Question #2 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

All Common Core: High School - Number and Quantity Resources

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