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Common Core: High School - Number and Quantity : Complex Solutions of Quadratic Equations: CCSS.Math.Content.HSN-CN.C.7

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Example Questions

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

Solve for $\displaystyle x$.

$\displaystyle 4x^2+4x+2$

Possible Answers:

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

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

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

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

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

Correct answer:

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

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.

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

where the quadratic function is in the form,

$\displaystyle ax^2+bx+c$.

Step 2: Identify the coefficients.

$\displaystyle 4x^2+4x+2$

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

Step 3: Substitute coefficients into the quadratic formula.

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

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

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

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

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

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

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

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

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.

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

Possible Answers:

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

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

$x=\pm 2$ $\displaystyle x=\pm 2$

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

Correct answer:

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

Explanation:

a=3 $\displaystyle a=3$

b=-2 $\displaystyle b=-2$

c=7 $\displaystyle c=7$

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

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

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

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

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

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

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Example Question #84 : High School: Number And Quantity

Solve.

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

Possible Answers:

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

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

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

$x=\pm 2i$ $\displaystyle x=\pm 2i$

Correct answer:

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

Explanation:

a=4 $\displaystyle a=4$

b=2 $\displaystyle b=2$

c=3 $\displaystyle c=3$

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

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

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

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

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

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

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

Solve.

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

Possible Answers:

$x=\pm 2i$ $\displaystyle x=\pm 2i$

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

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

$x=1\pm 2i$ $\displaystyle x=1\pm 2i$

Correct answer:

$x=1\pm 2i$ $\displaystyle x=1\pm 2i$

Explanation:

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

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

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

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

$x=1\pm 2i$ $\displaystyle x=1\pm 2i$

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

Solve.

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

Possible Answers:

$a=3\pm 2i$ $\displaystyle a=3\pm 2i$

$a=\pm 3$ $\displaystyle a=\pm 3$

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

$a=\pm 4i$ $\displaystyle a=\pm 4i$

Correct answer:

$a=3\pm 2i$ $\displaystyle a=3\pm 2i$

Explanation:

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

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

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

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

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

$a=3\pm 2i$ $\displaystyle a=3\pm 2i$

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

Solve.

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

Possible Answers:

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

$b=\pm 3$ $\displaystyle b=\pm 3$

$b=\pm 4i$ $\displaystyle b=\pm 4i$

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

Correct answer:

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

Explanation:

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

Solve.

$g^{2}-18g+84$ $\displaystyle g^{2}-18g+84$

Possible Answers:

$g=\pm i\sqrt{3}$ $\displaystyle g=\pm i\sqrt{3}$

$9\pm i\sqrt{3}$ $\displaystyle 9\pm i\sqrt{3}$

$g=\pm 4i$ $\displaystyle g=\pm 4i$

$g\pm 3$ $\displaystyle g\pm 3$

Correct answer:

$9\pm i\sqrt{3}$ $\displaystyle 9\pm i\sqrt{3}$

Explanation:

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Example Question #4 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Solve.

$2m^{2}-8m+25$ $\displaystyle 2m^{2}-8m+25$

Possible Answers:

$m=3\pm i\sqrt{17}$ $\displaystyle m=3\pm i\sqrt{17}$

$m=\frac{4\pm i\sqrt{17}}{2}$ $\displaystyle m=\frac{4\pm i\sqrt{17}}{2}$

$m=\pm 3i$ $\displaystyle m=\pm 3i$

$m=\pm 4$ $\displaystyle m=\pm 4$

Correct answer:

$m=\frac{4\pm i\sqrt{17}}{2}$ $\displaystyle m=\frac{4\pm i\sqrt{17}}{2}$

Explanation:

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

Solve.

$4a^{2}+25=0$ $\displaystyle 4a^{2}+25=0$

Possible Answers:

$a=\pm \frac{5}{2}$ $\displaystyle a=\pm \frac{5}{2}$

$a=\pm 5i$ $\displaystyle a=\pm 5i$

$a=\pm 25$ $\displaystyle a=\pm 25$

$a=\pm \frac{5}{2}i$ $\displaystyle a=\pm \frac{5}{2}i$

Correct answer:

$a=\pm \frac{5}{2}i$ $\displaystyle a=\pm \frac{5}{2}i$

Explanation:

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Example Question #6 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

Solve.

$8z^{2}+72=0$ $\displaystyle 8z^{2}+72=0$

Possible Answers:

$z=\pm 9i$ $\displaystyle z=\pm 9i$

$z=\pm 3$ $\displaystyle z=\pm 3$

$z=\pm 3i$ $\displaystyle z=\pm 3i$

$z=\sqrt{9}$ $\displaystyle z=\sqrt{9}$

Correct answer:

$z=\pm 3i$ $\displaystyle z=\pm 3i$

Explanation:

10.

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Common Core: High School - Number and Quantity : Complex Solutions of Quadratic Equations: CCSS.Math.Content.HSN-CN.C.7

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 #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 #84 : High School: Number And Quantity

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

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

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

Example Question #4 : Complex Solutions Of Quadratic Equations: Ccss.Math.Content.Hsn Cn.C.7

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

Example Question #6 : 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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