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Common Core: High School - Algebra : Solve Quadratic Equations by Inspection, Quadratic Formula, Factoring, Completing the Square, and Taking Square Roots: CCSS.Math.Content.HSA-REI.B.4b

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All Common Core: High School - Algebra Resources

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Example Questions

Example Question #11 : Solve Quadratic Equations By Inspection, Quadratic Formula, Factoring, Completing The Square, And Taking Square Roots: Ccss.Math.Content.Hsa Rei.B.4b

Solve $27 x^{2} + 3 x + 5 = 0$

Possible Answers:

$x = - \frac{37}{18} - \frac{\sqrt{59} i}{18} , x = \frac{359}{18} + \frac{\sqrt{59} i}{18}$

$x = - \frac{1}{9} - \frac{\sqrt{59} i}{9} , x = \frac{1}{9} - \frac{\sqrt{59} i}{9}$

$x = - \frac{1}{18} - \frac{\sqrt{59} i}{18} , x = - \frac{1}{18} + \frac{\sqrt{59} i}{18}$

$x = \frac{179}{18} - \frac{\sqrt{59} i}{18} , x = - \frac{181}{18} + \frac{\sqrt{59} i}{18}$

$x = \frac{17}{18} - \frac{\sqrt{59} i}{18} , x = - \frac{19}{18} + \frac{\sqrt{59} i}{18}$

Correct answer:

$x = - \frac{1}{18} - \frac{\sqrt{59} i}{18} , x = - \frac{1}{18} + \frac{\sqrt{59} i}{18}$

Explanation:

We can solve this by using the quadratic formula.

The quadratic formula is

$\frac{ -b \pm\sqrt{ b^{2} - 4 a c }}{ 2 a }$

$\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ correspond to coefficients in the quadratic equation, which is

$a x^{2} + b x + 5 = 0$

In this case a = 27 , b = 3 , and c = 5 .

$\frac{ -3 \pm\sqrt{ 3 ^2- 4 \cdot 27 \cdot 5 }}{ 2 \cdot 27 }$

$\frac{ -3 \pm\sqrt{ 9 - 540 }}{ 54 }$

$\frac{ -3 \pm\sqrt{ -531 }}{ 54 }$

Since the number inside the square root is negative, we will have an imaginary answer.

We will put an $\uptext{i}$ , outside the square root sign, and then do normal operations.

$\frac{ -3 \pm i \sqrt{ 531 }}{ 54 }$

Now we split this up into two equations.

$\frac{ -3 + i \sqrt{ 531 }}{ 54 }$

$- \frac{1}{18} + \frac{\sqrt{59} i}{18}$

$\frac{ -3 - i \sqrt{ 531 }}{ 54 }$

$- \frac{1}{18} - \frac{\sqrt{59} i}{18}$

So our solutions are $x = - \frac{1}{18} - \frac{\sqrt{59} i}{18}$ and $x = - \frac{1}{18} + \frac{\sqrt{59} i}{18}$

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Example Question #61 : Reasoning With Equations & Inequalities

Solve $12 x^{2} - 13 x + 41 = 0$

Possible Answers:

$x = \frac{37}{24} - \frac{\sqrt{1799} i}{24} , x = - \frac{11}{24} + \frac{\sqrt{1799} i}{24}$

$x = - \frac{35}{24} - \frac{\sqrt{1799} i}{24} , x = \frac{493}{24} + \frac{\sqrt{1799} i}{24}$

$x = \frac{13}{24} - \frac{\sqrt{1799} i}{24} , x = \frac{13}{24} + \frac{\sqrt{1799} i}{24}$

$x = \frac{253}{24} - \frac{\sqrt{1799} i}{24} , x = - \frac{227}{24} + \frac{\sqrt{1799} i}{24}$

$x = \frac{13}{12} - \frac{\sqrt{1799} i}{12} , x = - \frac{13}{12} - \frac{\sqrt{1799} i}{12}$

Correct answer:

$x = \frac{13}{24} - \frac{\sqrt{1799} i}{24} , x = \frac{13}{24} + \frac{\sqrt{1799} i}{24}$

Explanation:

We can solve this by using the quadratic formula.

The quadratic formula is

$\frac{ -b \pm\sqrt{ b^{2} - 4 a c }}{ 2 a }$

$\uptext{a}$ , $\uptext{b}$ , and $\uptext{c}$ correspond to coefficients in the quadratic equation, which is

$a x^{2} + b x + 41 = 0$

In this case a = 12 , b = -13 , and c = 41 .

$\frac{ 13 \pm\sqrt{ -13 ^2- 4 \cdot 12 \cdot 41 }}{ 2 \cdot 12 }$

$\frac{ 13 \pm\sqrt{ 169 - 1968 }}{ 24 }$

$\frac{ 13 \pm\sqrt{ -1799 }}{ 24 }$

Since the number inside the square root is negative, we will have an imaginary answer.

We will put an $\uptext{i}$ , outside the square root sign, and then do normal operations.

$\frac{ 13 \pm i \sqrt{ 1799 }}{ 24 }$

Now we split this up into two equations.

$\frac{ 13 + i \sqrt{ 1799 }}{ 24 }$

$\frac{13}{24} + \frac{\sqrt{1799} i}{24}$

$\frac{ 13 - i \sqrt{ 1799 }}{ 24 }$

$\frac{13}{24} - \frac{\sqrt{1799} i}{24}$

So our solutions are $x = \frac{13}{24} - \frac{\sqrt{1799} i}{24}$ and $x = \frac{13}{24} + \frac{\sqrt{1799} i}{24}$

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Common Core: High School - Algebra : Solve Quadratic Equations by Inspection, Quadratic Formula, Factoring, Completing the Square, and Taking Square Roots: CCSS.Math.Content.HSA-REI.B.4b

Study concepts, example questions & explanations for Common Core: High School - Algebra

All Common Core: High School - Algebra Resources

Example Questions

Example Question #11 : Solve Quadratic Equations By Inspection, Quadratic Formula, Factoring, Completing The Square, And Taking Square Roots: Ccss.Math.Content.Hsa Rei.B.4b

Example Question #61 : Reasoning With Equations & Inequalities

All Common Core: High School - Algebra Resources

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