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

Solve $- 11 x^{2} + 20 x - 44 = 0$

$x = \frac{10}{11} - \frac{8 i}{11} \sqrt{6} , x = \frac{10}{11} + \frac{8 i}{11} \sqrt{6}$

$x = \frac{21}{11} - \frac{8 i}{11} \sqrt{6} , x = - \frac{1}{11} + \frac{8 i}{11} \sqrt{6}$

$x = \frac{120}{11} - \frac{8 i}{11} \sqrt{6} , x = - \frac{100}{11} + \frac{8 i}{11} \sqrt{6}$

$x = - \frac{12}{11} - \frac{8 i}{11} \sqrt{6} , x = \frac{230}{11} + \frac{8 i}{11} \sqrt{6}$

$x = \frac{20}{11} - \frac{16 i}{11} \sqrt{6} , x = - \frac{20}{11} - \frac{16 i}{11} \sqrt{6}$

Explanation

We can solve this by using the quadratic formula.

The quadratic formula is

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

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

$a x^{2} + b x - 44 = 0$

In this case , , and .

$\frac{ -20 \pm\sqrt{ 20 ^2- 4 \cdot -11 \cdot -44 }}{ 2 \cdot -11 }$

$\frac{ -20 \pm\sqrt{ 400 - 1936 }}{ -22 }$

$\frac{ -20 \pm\sqrt{ -1536 }}{ -22 }$

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.

$\frac{ -20 \pm i \sqrt{ 1536 }}{ -22 }$

Now we split this up into two equations.

$\frac{ -20 + i \sqrt{ 1536 }}{ -22 }$

$\frac{10}{11} + \frac{8 i}{11} \sqrt{6}$

$\frac{ -20 - i \sqrt{ 1536 }}{ -22 }$

$\frac{10}{11} - \frac{8 i}{11} \sqrt{6}$

So our solutions are $x = \frac{10}{11} - \frac{8 i}{11} \sqrt{6}$ and $x = \frac{10}{11} + \frac{8 i}{11} \sqrt{6}$

Solve $- 24 x^{2} + 17 x - 32 = 0$

$x = \frac{17}{48} - \frac{11 i}{48} \sqrt{23} , x = \frac{17}{48} + \frac{11 i}{48} \sqrt{23}$

$x = \frac{65}{48} - \frac{11 i}{48} \sqrt{23} , x = - \frac{31}{48} + \frac{11 i}{48} \sqrt{23}$

$x = \frac{497}{48} - \frac{11 i}{48} \sqrt{23} , x = - \frac{463}{48} + \frac{11 i}{48} \sqrt{23}$

$x = - \frac{79}{48} - \frac{11 i}{48} \sqrt{23} , x = \frac{977}{48} + \frac{11 i}{48} \sqrt{23}$

$x = \frac{17}{24} - \frac{11 i}{24} \sqrt{23} , x = - \frac{17}{24} - \frac{11 i}{24} \sqrt{23}$

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

$a x^{2} + b x - 32 = 0$

In this case , , and .

$\frac{ -17 \pm\sqrt{ 17 ^2- 4 \cdot -24 \cdot -32 }}{ 2 \cdot -24 }$

$\frac{ -17 \pm\sqrt{ 289 - 3072 }}{ -48 }$

$\frac{ -17 \pm\sqrt{ -2783 }}{ -48 }$

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.

$\frac{ -17 \pm i \sqrt{ 2783 }}{ -48 }$

Now we split this up into two equations.

$\frac{ -17 + i \sqrt{ 2783 }}{ -48 }$

$\frac{17}{48} + \frac{11 i}{48} \sqrt{23}$

$\frac{ -17 - i \sqrt{ 2783 }}{ -48 }$

$\frac{17}{48} - \frac{11 i}{48} \sqrt{23}$

So our solutions are $x = \frac{17}{48} - \frac{11 i}{48} \sqrt{23}$ and $x = \frac{17}{48} + \frac{11 i}{48} \sqrt{23}$

Solve $- 23 x^{2} + 34 x - 44 = 0$

$x = \frac{17}{23} - \frac{\sqrt{723} i}{23} , x = \frac{17}{23} + \frac{\sqrt{723} i}{23}$

$x = \frac{40}{23} - \frac{\sqrt{723} i}{23} , x = - \frac{6}{23} + \frac{\sqrt{723} i}{23}$

$x = \frac{247}{23} - \frac{\sqrt{723} i}{23} , x = - \frac{213}{23} + \frac{\sqrt{723} i}{23}$

$x = - \frac{29}{23} - \frac{\sqrt{723} i}{23} , x = \frac{477}{23} + \frac{\sqrt{723} i}{23}$

$x = \frac{34}{23} - \frac{2 i}{23} \sqrt{723} , x = - \frac{34}{23} - \frac{2 i}{23} \sqrt{723}$

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

$a x^{2} + b x - 44 = 0$

In this case , , and .

$\frac{ -34 \pm\sqrt{ 34 ^2- 4 \cdot -23 \cdot -44 }}{ 2 \cdot -23 }$

$\frac{ -34 \pm\sqrt{ 1156 - 4048 }}{ -46 }$

$\frac{ -34 \pm\sqrt{ -2892 }}{ -46 }$

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.

$\frac{ -34 \pm i \sqrt{ 2892 }}{ -46 }$

Now we split this up into two equations.

$\frac{ -34 + i \sqrt{ 2892 }}{ -46 }$

$\frac{17}{23} + \frac{\sqrt{723} i}{23}$

$\frac{ -34 - i \sqrt{ 2892 }}{ -46 }$

$\frac{17}{23} - \frac{\sqrt{723} i}{23}$

So our solutions are $x = \frac{17}{23} - \frac{\sqrt{723} i}{23}$ and $x = \frac{17}{23} + \frac{\sqrt{723} i}{23}$

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

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

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

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

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

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

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 , , and .

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

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

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

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

Solve $18 x^{2} - 14 x + 41 = 0$

$x = \frac{7}{18} - \frac{\sqrt{689} i}{18} , x = \frac{7}{18} + \frac{\sqrt{689} i}{18}$

$x = \frac{25}{18} - \frac{\sqrt{689} i}{18} , x = - \frac{11}{18} + \frac{\sqrt{689} i}{18}$

$x = \frac{187}{18} - \frac{\sqrt{689} i}{18} , x = - \frac{173}{18} + \frac{\sqrt{689} i}{18}$

$x = - \frac{29}{18} - \frac{\sqrt{689} i}{18} , x = \frac{367}{18} + \frac{\sqrt{689} i}{18}$

$x = \frac{7}{9} - \frac{\sqrt{689} i}{9} , x = - \frac{7}{9} - \frac{\sqrt{689} i}{9}$

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 , , and .

$\frac{ 14 \pm\sqrt{ -14 ^2- 4 \cdot 18 \cdot 41 }}{ 2 \cdot 18 }$

$\frac{ 14 \pm\sqrt{ 196 - 2952 }}{ 36 }$

$\frac{ 14 \pm\sqrt{ -2756 }}{ 36 }$

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.

$\frac{ 14 \pm i \sqrt{ 2756 }}{ 36 }$

Now we split this up into two equations.

$\frac{ 14 + i \sqrt{ 2756 }}{ 36 }$

$\frac{7}{18} + \frac{\sqrt{689} i}{18}$

$\frac{ 14 - i \sqrt{ 2756 }}{ 36 }$

$\frac{7}{18} - \frac{\sqrt{689} i}{18}$

So our solutions are $x = \frac{7}{18} - \frac{\sqrt{689} i}{18}$ and $x = \frac{7}{18} + \frac{\sqrt{689} i}{18}$

Solve $- 49 x^{2} + 17 x - 45 = 0$

$x = \frac{17}{98} - \frac{\sqrt{8531} i}{98} , x = \frac{17}{98} + \frac{\sqrt{8531} i}{98}$

$x = \frac{115}{98} - \frac{\sqrt{8531} i}{98} , x = - \frac{81}{98} + \frac{\sqrt{8531} i}{98}$

$x = \frac{997}{98} - \frac{\sqrt{8531} i}{98} , x = - \frac{963}{98} + \frac{\sqrt{8531} i}{98}$

$x = - \frac{179}{98} - \frac{\sqrt{8531} i}{98} , x = \frac{1977}{98} + \frac{\sqrt{8531} i}{98}$

$x = \frac{17}{49} - \frac{\sqrt{8531} i}{49} , x = - \frac{17}{49} - \frac{\sqrt{8531} i}{49}$

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

$a x^{2} + b x - 45 = 0$

In this case , , and .

$\frac{ -17 \pm\sqrt{ 17 ^2- 4 \cdot -49 \cdot -45 }}{ 2 \cdot -49 }$

$\frac{ -17 \pm\sqrt{ 289 - 8820 }}{ -98 }$

$\frac{ -17 \pm\sqrt{ -8531 }}{ -98 }$

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.

$\frac{ -17 \pm i \sqrt{ 8531 }}{ -98 }$

Now we split this up into two equations.

$\frac{ -17 + i \sqrt{ 8531 }}{ -98 }$

$\frac{17}{98} + \frac{\sqrt{8531} i}{98}$

$\frac{ -17 - i \sqrt{ 8531 }}{ -98 }$

$\frac{17}{98} - \frac{\sqrt{8531} i}{98}$

So our solutions are $x = \frac{17}{98} - \frac{\sqrt{8531} i}{98}$ and $x = \frac{17}{98} + \frac{\sqrt{8531} i}{98}$

Solve $- 49 x^{2} + 24 x - 40 = 0$

$x = \frac{12}{49} - \frac{2 i}{49} \sqrt{454} , x = \frac{12}{49} + \frac{2 i}{49} \sqrt{454}$

$x = \frac{61}{49} - \frac{2 i}{49} \sqrt{454} , x = - \frac{37}{49} + \frac{2 i}{49} \sqrt{454}$

$x = \frac{502}{49} - \frac{2 i}{49} \sqrt{454} , x = - \frac{478}{49} + \frac{2 i}{49} \sqrt{454}$

$x = - \frac{86}{49} - \frac{2 i}{49} \sqrt{454} , x = \frac{992}{49} + \frac{2 i}{49} \sqrt{454}$

$x = \frac{24}{49} - \frac{4 i}{49} \sqrt{454} , x = - \frac{24}{49} - \frac{4 i}{49} \sqrt{454}$

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

$a x^{2} + b x - 40 = 0$

In this case , , and .

$\frac{ -24 \pm\sqrt{ 24 ^2- 4 \cdot -49 \cdot -40 }}{ 2 \cdot -49 }$

$\frac{ -24 \pm\sqrt{ 576 - 7840 }}{ -98 }$

$\frac{ -24 \pm\sqrt{ -7264 }}{ -98 }$

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.

$\frac{ -24 \pm i \sqrt{ 7264 }}{ -98 }$

Now we split this up into two equations.

$\frac{ -24 + i \sqrt{ 7264 }}{ -98 }$

$\frac{12}{49} + \frac{2 i}{49} \sqrt{454}$

$\frac{ -24 - i \sqrt{ 7264 }}{ -98 }$

$\frac{12}{49} - \frac{2 i}{49} \sqrt{454}$

So our solutions are $x = \frac{12}{49} - \frac{2 i}{49} \sqrt{454}$ and $x = \frac{12}{49} + \frac{2 i}{49} \sqrt{454}$

Solve $- 41 x^{2} + 46 x - 20 = 0$

$x = \frac{23}{41} - \frac{\sqrt{291} i}{41} , x = \frac{23}{41} + \frac{\sqrt{291} i}{41}$

$x = \frac{64}{41} - \frac{\sqrt{291} i}{41} , x = - \frac{18}{41} + \frac{\sqrt{291} i}{41}$

$x = \frac{433}{41} - \frac{\sqrt{291} i}{41} , x = - \frac{387}{41} + \frac{\sqrt{291} i}{41}$

$x = - \frac{59}{41} - \frac{\sqrt{291} i}{41} , x = \frac{843}{41} + \frac{\sqrt{291} i}{41}$

$x = \frac{46}{41} - \frac{2 i}{41} \sqrt{291} , x = - \frac{46}{41} - \frac{2 i}{41} \sqrt{291}$

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

$a x^{2} + b x - 20 = 0$

In this case , , and .

$\frac{ -46 \pm\sqrt{ 46 ^2- 4 \cdot -41 \cdot -20 }}{ 2 \cdot -41 }$

$\frac{ -46 \pm\sqrt{ 2116 - 3280 }}{ -82 }$

$\frac{ -46 \pm\sqrt{ -1164 }}{ -82 }$

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.

$\frac{ -46 \pm i \sqrt{ 1164 }}{ -82 }$

Now we split this up into two equations.

$\frac{ -46 + i \sqrt{ 1164 }}{ -82 }$

$\frac{23}{41} + \frac{\sqrt{291} i}{41}$

$\frac{ -46 - i \sqrt{ 1164 }}{ -82 }$

$\frac{23}{41} - \frac{\sqrt{291} i}{41}$

So our solutions are $x = \frac{23}{41} - \frac{\sqrt{291} i}{41}$ and $x = \frac{23}{41} + \frac{\sqrt{291} i}{41}$

Solve $22 x^{2} - 9 x + 16 = 0$

$x = \frac{9}{44} - \frac{\sqrt{1327} i}{44} , x = \frac{9}{44} + \frac{\sqrt{1327} i}{44}$

$x = \frac{53}{44} - \frac{\sqrt{1327} i}{44} , x = - \frac{35}{44} + \frac{\sqrt{1327} i}{44}$

$x = \frac{449}{44} - \frac{\sqrt{1327} i}{44} , x = - \frac{431}{44} + \frac{\sqrt{1327} i}{44}$

$x = - \frac{79}{44} - \frac{\sqrt{1327} i}{44} , x = \frac{889}{44} + \frac{\sqrt{1327} i}{44}$

$x = \frac{9}{22} - \frac{\sqrt{1327} i}{22} , x = - \frac{9}{22} - \frac{\sqrt{1327} i}{22}$

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

$a x^{2} + b x + 16 = 0$

In this case , , and .

$\frac{ 9 \pm\sqrt{ -9 ^2- 4 \cdot 22 \cdot 16 }}{ 2 \cdot 22 }$

$\frac{ 9 \pm\sqrt{ 81 - 1408 }}{ 44 }$

$\frac{ 9 \pm\sqrt{ -1327 }}{ 44 }$

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.

$\frac{ 9 \pm i \sqrt{ 1327 }}{ 44 }$

Now we split this up into two equations.

$\frac{ 9 + i \sqrt{ 1327 }}{ 44 }$

$\frac{9}{44} + \frac{\sqrt{1327} i}{44}$

$\frac{ 9 - i \sqrt{ 1327 }}{ 44 }$

$\frac{9}{44} - \frac{\sqrt{1327} i}{44}$

So our solutions are $x = \frac{9}{44} - \frac{\sqrt{1327} i}{44}$ and $x = \frac{9}{44} + \frac{\sqrt{1327} i}{44}$

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

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

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

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

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

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

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 , , and .

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

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

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

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

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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