Pre-Algebra Examples

$3 x^{2} + 4 y^{2} - 6 x - 40 y + 103 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 4$ , $b = - 40$ , and $c = 3 x^{2} - 6 x + 103$ into the quadratic formula and solve for $y$ .

$\frac{40 \pm \sqrt{{(- 40)}^{2} - 4 \cdot (4 \cdot (3 x^{2} - 6 x + 103))}}{2 \cdot 4}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 40$ to the power of $2$ .

$y = \frac{40 \pm \sqrt{1600 - 4 \cdot 4 \cdot (3 x^{2} - 6 x + 103)}}{2 \cdot 4}$

Step 3.1.2

Multiply $- 4$ by $4$ .

$y = \frac{40 \pm \sqrt{1600 - 16 \cdot (3 x^{2} - 6 x + 103)}}{2 \cdot 4}$

Step 3.1.3

Apply the distributive property.

$y = \frac{40 \pm \sqrt{1600 - 16 (3 x^{2}) - 16 (- 6 x) - 16 \cdot 103}}{2 \cdot 4}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $3$ by $- 16$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} - 16 (- 6 x) - 16 \cdot 103}}{2 \cdot 4}$

Step 3.1.4.2

Multiply $- 6$ by $- 16$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} + 96 x - 16 \cdot 103}}{2 \cdot 4}$

Step 3.1.4.3

Multiply $- 16$ by $103$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} + 96 x - 1648}}{2 \cdot 4}$

Step 3.1.5

Subtract $1648$ from $1600$ .

$y = \frac{40 \pm \sqrt{- 48 x^{2} + 96 x - 48}}{2 \cdot 4}$

Step 3.1.6

Rewrite $- 48 x^{2} + 96 x - 48$ in a factored form.

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Step 3.1.6.1

Factor $48$ out of $- 48 x^{2} + 96 x - 48$ .

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Step 3.1.6.1.1

Factor $48$ out of $- 48 x^{2}$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 96 x - 48}}{2 \cdot 4}$

Step 3.1.6.1.2

Factor $48$ out of $96 x$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 48 (2 x) - 48}}{2 \cdot 4}$

Step 3.1.6.1.3

Factor $48$ out of $- 48$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 48 (2 x) + 48 (- 1)}}{2 \cdot 4}$

Step 3.1.6.1.4

Factor $48$ out of $48 (- x^{2}) + 48 (2 x)$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 x) + 48 (- 1)}}{2 \cdot 4}$

Step 3.1.6.1.5

Factor $48$ out of $48 (- x^{2} + 2 x) + 48 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 x - 1)}}{2 \cdot 4}$

Step 3.1.6.2

Factor by grouping.

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Step 3.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 3.1.6.2.1.1

Factor $2$ out of $2 x$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 (x) - 1)}}{2 \cdot 4}$

Step 3.1.6.2.1.2

Rewrite $2$ as $1$ plus $1$

$y = \frac{40 \pm \sqrt{48 (- x^{2} + (1 + 1) x - 1)}}{2 \cdot 4}$

Step 3.1.6.2.1.3

Apply the distributive property.

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 1 x + 1 x - 1)}}{2 \cdot 4}$

Step 3.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + x + 1 x - 1)}}{2 \cdot 4}$

Step 3.1.6.2.1.5

Multiply $x$ by $1$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + x + x - 1)}}{2 \cdot 4}$

Step 3.1.6.2.2

Factor out the greatest common factor from each group.

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Step 3.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{40 \pm \sqrt{48 ((- x^{2} + x) + x - 1)}}{2 \cdot 4}$

Step 3.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{40 \pm \sqrt{48 (x (- x + 1) - 1 (- x + 1))}}{2 \cdot 4}$

Step 3.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{40 \pm \sqrt{48 ((- x + 1) (x - 1))}}{2 \cdot 4}$

$y = \frac{40 \pm \sqrt{48 (- x + 1) (x - 1)}}{2 \cdot 4}$

Step 3.1.6.3

Combine exponents.

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Step 3.1.6.3.1

Factor $- 1$ out of $- x$ .

$y = \frac{40 \pm \sqrt{48 (- (x) + 1) (x - 1)}}{2 \cdot 4}$

Step 3.1.6.3.2

Rewrite $1$ as $- 1 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- (x) - 1 \cdot - 1) (x - 1)}}{2 \cdot 4}$

Step 3.1.6.3.3

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- (x - 1)) (x - 1)}}{2 \cdot 4}$

Step 3.1.6.3.4

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$y = \frac{40 \pm \sqrt{48 (- 1 (x - 1)) (x - 1)}}{2 \cdot 4}$

Step 3.1.6.3.5

Raise $x - 1$ to the power of $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 ((x - 1) (x - 1)))}}{2 \cdot 4}$

Step 3.1.6.3.6

Raise $x - 1$ to the power of $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 ((x - 1) (x - 1)))}}{2 \cdot 4}$

Step 3.1.6.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 {(x - 1)}^{1 + 1})}}{2 \cdot 4}$

Step 3.1.6.3.8

Add $1$ and $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 {(x - 1)}^{2})}}{2 \cdot 4}$

Step 3.1.6.3.9

Multiply $48$ by $- 1$ .

$y = \frac{40 \pm \sqrt{- 48 {(x - 1)}^{2}}}{2 \cdot 4}$

Step 3.1.7

Rewrite $- 48 {(x - 1)}^{2}$ as ${(4 (x - 1))}^{2} \cdot - 3$ .

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Step 3.1.7.1

Factor $16$ out of $- 48$ .

$y = \frac{40 \pm \sqrt{16 (- 3) {(x - 1)}^{2}}}{2 \cdot 4}$

Step 3.1.7.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{40 \pm \sqrt{4^{2} \cdot (- 3 {(x - 1)}^{2})}}{2 \cdot 4}$

Step 3.1.7.3

Move $- 3$ .

$y = \frac{40 \pm \sqrt{4^{2} {(x - 1)}^{2} \cdot - 3}}{2 \cdot 4}$

Step 3.1.7.4

Rewrite $4^{2} {(x - 1)}^{2}$ as ${(4 (x - 1))}^{2}$ .

$y = \frac{40 \pm \sqrt{{(4 (x - 1))}^{2} \cdot - 3}}{2 \cdot 4}$

Step 3.1.8

Pull terms out from under the radical.

$y = \frac{40 \pm 4 (x - 1) \sqrt{- 3}}{2 \cdot 4}$

Step 3.1.9

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{40 \pm 4 (x - 1) \sqrt{- 1 \cdot 3}}{2 \cdot 4}$

Step 3.1.10

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$y = \frac{40 \pm 4 (x - 1) (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot 4}$

Step 3.1.11

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{40 \pm 4 (x - 1) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 3.1.12

Apply the distributive property.

$y = \frac{40 \pm (4 x + 4 \cdot - 1) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 3.1.13

Multiply $4$ by $- 1$ .

$y = \frac{40 \pm (4 x - 4) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 3.1.14

Apply the distributive property.

$y = \frac{40 \pm (4 x i \sqrt{3} - 4 i \sqrt{3})}{2 \cdot 4}$

Step 3.2

Multiply $2$ by $4$ .

$y = \frac{40 \pm (4 x i \sqrt{3} - 4 i \sqrt{3})}{8}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 40$ to the power of $2$ .

$y = \frac{40 \pm \sqrt{1600 - 4 \cdot 4 \cdot (3 x^{2} - 6 x + 103)}}{2 \cdot 4}$

Step 4.1.2

Multiply $- 4$ by $4$ .

$y = \frac{40 \pm \sqrt{1600 - 16 \cdot (3 x^{2} - 6 x + 103)}}{2 \cdot 4}$

Step 4.1.3

Apply the distributive property.

$y = \frac{40 \pm \sqrt{1600 - 16 (3 x^{2}) - 16 (- 6 x) - 16 \cdot 103}}{2 \cdot 4}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $3$ by $- 16$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} - 16 (- 6 x) - 16 \cdot 103}}{2 \cdot 4}$

Step 4.1.4.2

Multiply $- 6$ by $- 16$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} + 96 x - 16 \cdot 103}}{2 \cdot 4}$

Step 4.1.4.3

Multiply $- 16$ by $103$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} + 96 x - 1648}}{2 \cdot 4}$

Step 4.1.5

Subtract $1648$ from $1600$ .

$y = \frac{40 \pm \sqrt{- 48 x^{2} + 96 x - 48}}{2 \cdot 4}$

Step 4.1.6

Rewrite $- 48 x^{2} + 96 x - 48$ in a factored form.

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Step 4.1.6.1

Factor $48$ out of $- 48 x^{2} + 96 x - 48$ .

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Step 4.1.6.1.1

Factor $48$ out of $- 48 x^{2}$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 96 x - 48}}{2 \cdot 4}$

Step 4.1.6.1.2

Factor $48$ out of $96 x$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 48 (2 x) - 48}}{2 \cdot 4}$

Step 4.1.6.1.3

Factor $48$ out of $- 48$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 48 (2 x) + 48 (- 1)}}{2 \cdot 4}$

Step 4.1.6.1.4

Factor $48$ out of $48 (- x^{2}) + 48 (2 x)$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 x) + 48 (- 1)}}{2 \cdot 4}$

Step 4.1.6.1.5

Factor $48$ out of $48 (- x^{2} + 2 x) + 48 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 x - 1)}}{2 \cdot 4}$

Step 4.1.6.2

Factor by grouping.

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Step 4.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 4.1.6.2.1.1

Factor $2$ out of $2 x$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 (x) - 1)}}{2 \cdot 4}$

Step 4.1.6.2.1.2

Rewrite $2$ as $1$ plus $1$

$y = \frac{40 \pm \sqrt{48 (- x^{2} + (1 + 1) x - 1)}}{2 \cdot 4}$

Step 4.1.6.2.1.3

Apply the distributive property.

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 1 x + 1 x - 1)}}{2 \cdot 4}$

Step 4.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + x + 1 x - 1)}}{2 \cdot 4}$

Step 4.1.6.2.1.5

Multiply $x$ by $1$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + x + x - 1)}}{2 \cdot 4}$

Step 4.1.6.2.2

Factor out the greatest common factor from each group.

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Step 4.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{40 \pm \sqrt{48 ((- x^{2} + x) + x - 1)}}{2 \cdot 4}$

Step 4.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{40 \pm \sqrt{48 (x (- x + 1) - 1 (- x + 1))}}{2 \cdot 4}$

Step 4.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{40 \pm \sqrt{48 ((- x + 1) (x - 1))}}{2 \cdot 4}$

$y = \frac{40 \pm \sqrt{48 (- x + 1) (x - 1)}}{2 \cdot 4}$

Step 4.1.6.3

Combine exponents.

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Step 4.1.6.3.1

Factor $- 1$ out of $- x$ .

$y = \frac{40 \pm \sqrt{48 (- (x) + 1) (x - 1)}}{2 \cdot 4}$

Step 4.1.6.3.2

Rewrite $1$ as $- 1 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- (x) - 1 \cdot - 1) (x - 1)}}{2 \cdot 4}$

Step 4.1.6.3.3

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- (x - 1)) (x - 1)}}{2 \cdot 4}$

Step 4.1.6.3.4

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$y = \frac{40 \pm \sqrt{48 (- 1 (x - 1)) (x - 1)}}{2 \cdot 4}$

Step 4.1.6.3.5

Raise $x - 1$ to the power of $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 ((x - 1) (x - 1)))}}{2 \cdot 4}$

Step 4.1.6.3.6

Raise $x - 1$ to the power of $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 ((x - 1) (x - 1)))}}{2 \cdot 4}$

Step 4.1.6.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 {(x - 1)}^{1 + 1})}}{2 \cdot 4}$

Step 4.1.6.3.8

Add $1$ and $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 {(x - 1)}^{2})}}{2 \cdot 4}$

Step 4.1.6.3.9

Multiply $48$ by $- 1$ .

$y = \frac{40 \pm \sqrt{- 48 {(x - 1)}^{2}}}{2 \cdot 4}$

Step 4.1.7

Rewrite $- 48 {(x - 1)}^{2}$ as ${(4 (x - 1))}^{2} \cdot - 3$ .

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Step 4.1.7.1

Factor $16$ out of $- 48$ .

$y = \frac{40 \pm \sqrt{16 (- 3) {(x - 1)}^{2}}}{2 \cdot 4}$

Step 4.1.7.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{40 \pm \sqrt{4^{2} \cdot (- 3 {(x - 1)}^{2})}}{2 \cdot 4}$

Step 4.1.7.3

Move $- 3$ .

$y = \frac{40 \pm \sqrt{4^{2} {(x - 1)}^{2} \cdot - 3}}{2 \cdot 4}$

Step 4.1.7.4

Rewrite $4^{2} {(x - 1)}^{2}$ as ${(4 (x - 1))}^{2}$ .

$y = \frac{40 \pm \sqrt{{(4 (x - 1))}^{2} \cdot - 3}}{2 \cdot 4}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{40 \pm 4 (x - 1) \sqrt{- 3}}{2 \cdot 4}$

Step 4.1.9

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{40 \pm 4 (x - 1) \sqrt{- 1 \cdot 3}}{2 \cdot 4}$

Step 4.1.10

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$y = \frac{40 \pm 4 (x - 1) (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot 4}$

Step 4.1.11

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{40 \pm 4 (x - 1) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 4.1.12

Apply the distributive property.

$y = \frac{40 \pm (4 x + 4 \cdot - 1) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 4.1.13

Multiply $4$ by $- 1$ .

$y = \frac{40 \pm (4 x - 4) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 4.1.14

Apply the distributive property.

$y = \frac{40 \pm (4 x i \sqrt{3} - 4 i \sqrt{3})}{2 \cdot 4}$

Step 4.2

Multiply $2$ by $4$ .

$y = \frac{40 \pm (4 x i \sqrt{3} - 4 i \sqrt{3})}{8}$

Step 4.3

Change the $\pm$ to $+$ .

$y = \frac{40 + 4 x i \sqrt{3} - 4 i \sqrt{3}}{8}$

Step 4.4

Cancel the common factor of $40 + 4 x i \sqrt{3} - 4 i \sqrt{3}$ and $8$ .

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Step 4.4.1

Factor $4$ out of $40$ .

$y = \frac{4 (10) + 4 x i \sqrt{3} - 4 i \sqrt{3}}{8}$

Step 4.4.2

Factor $4$ out of $4 x i \sqrt{3}$ .

$y = \frac{4 (10) + 4 (x i \sqrt{3}) - 4 i \sqrt{3}}{8}$

Step 4.4.3

Factor $4$ out of $- 4 i \sqrt{3}$ .

$y = \frac{4 (10) + 4 (x i \sqrt{3}) + 4 (- i \sqrt{3})}{8}$

Step 4.4.4

Factor $4$ out of $4 (x i \sqrt{3}) + 4 (- i \sqrt{3})$ .

$y = \frac{4 (10) + 4 (x i \sqrt{3} - i \sqrt{3})}{8}$

Step 4.4.5

Factor $4$ out of $4 (10) + 4 (x i \sqrt{3} - i \sqrt{3})$ .

$y = \frac{4 (10 + x i \sqrt{3} - i \sqrt{3})}{8}$

Step 4.4.6

Cancel the common factors.

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Step 4.4.6.1

Factor $4$ out of $8$ .

$y = \frac{4 (10 + x i \sqrt{3} - i \sqrt{3})}{4 \cdot 2}$

Step 4.4.6.2

Cancel the common factor.

$y = \frac{4 (10 + x i \sqrt{3} - i \sqrt{3})}{4 \cdot 2}$

Step 4.4.6.3

Rewrite the expression.

$y = \frac{10 + x i \sqrt{3} - i \sqrt{3}}{2}$

Step 4.5

Reorder terms.

$y = \frac{i x \sqrt{3} - i \sqrt{3} + 10}{2}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 40$ to the power of $2$ .

$y = \frac{40 \pm \sqrt{1600 - 4 \cdot 4 \cdot (3 x^{2} - 6 x + 103)}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4$ by $4$ .

$y = \frac{40 \pm \sqrt{1600 - 16 \cdot (3 x^{2} - 6 x + 103)}}{2 \cdot 4}$

Step 5.1.3

Apply the distributive property.

$y = \frac{40 \pm \sqrt{1600 - 16 (3 x^{2}) - 16 (- 6 x) - 16 \cdot 103}}{2 \cdot 4}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $3$ by $- 16$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} - 16 (- 6 x) - 16 \cdot 103}}{2 \cdot 4}$

Step 5.1.4.2

Multiply $- 6$ by $- 16$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} + 96 x - 16 \cdot 103}}{2 \cdot 4}$

Step 5.1.4.3

Multiply $- 16$ by $103$ .

$y = \frac{40 \pm \sqrt{1600 - 48 x^{2} + 96 x - 1648}}{2 \cdot 4}$

Step 5.1.5

Subtract $1648$ from $1600$ .

$y = \frac{40 \pm \sqrt{- 48 x^{2} + 96 x - 48}}{2 \cdot 4}$

Step 5.1.6

Rewrite $- 48 x^{2} + 96 x - 48$ in a factored form.

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Step 5.1.6.1

Factor $48$ out of $- 48 x^{2} + 96 x - 48$ .

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Step 5.1.6.1.1

Factor $48$ out of $- 48 x^{2}$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 96 x - 48}}{2 \cdot 4}$

Step 5.1.6.1.2

Factor $48$ out of $96 x$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 48 (2 x) - 48}}{2 \cdot 4}$

Step 5.1.6.1.3

Factor $48$ out of $- 48$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2}) + 48 (2 x) + 48 (- 1)}}{2 \cdot 4}$

Step 5.1.6.1.4

Factor $48$ out of $48 (- x^{2}) + 48 (2 x)$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 x) + 48 (- 1)}}{2 \cdot 4}$

Step 5.1.6.1.5

Factor $48$ out of $48 (- x^{2} + 2 x) + 48 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 x - 1)}}{2 \cdot 4}$

Step 5.1.6.2

Factor by grouping.

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Step 5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 5.1.6.2.1.1

Factor $2$ out of $2 x$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 2 (x) - 1)}}{2 \cdot 4}$

Step 5.1.6.2.1.2

Rewrite $2$ as $1$ plus $1$

$y = \frac{40 \pm \sqrt{48 (- x^{2} + (1 + 1) x - 1)}}{2 \cdot 4}$

Step 5.1.6.2.1.3

Apply the distributive property.

$y = \frac{40 \pm \sqrt{48 (- x^{2} + 1 x + 1 x - 1)}}{2 \cdot 4}$

Step 5.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + x + 1 x - 1)}}{2 \cdot 4}$

Step 5.1.6.2.1.5

Multiply $x$ by $1$ .

$y = \frac{40 \pm \sqrt{48 (- x^{2} + x + x - 1)}}{2 \cdot 4}$

Step 5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 5.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{40 \pm \sqrt{48 ((- x^{2} + x) + x - 1)}}{2 \cdot 4}$

Step 5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{40 \pm \sqrt{48 (x (- x + 1) - 1 (- x + 1))}}{2 \cdot 4}$

Step 5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{40 \pm \sqrt{48 ((- x + 1) (x - 1))}}{2 \cdot 4}$

$y = \frac{40 \pm \sqrt{48 (- x + 1) (x - 1)}}{2 \cdot 4}$

Step 5.1.6.3

Combine exponents.

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Step 5.1.6.3.1

Factor $- 1$ out of $- x$ .

$y = \frac{40 \pm \sqrt{48 (- (x) + 1) (x - 1)}}{2 \cdot 4}$

Step 5.1.6.3.2

Rewrite $1$ as $- 1 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- (x) - 1 \cdot - 1) (x - 1)}}{2 \cdot 4}$

Step 5.1.6.3.3

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$y = \frac{40 \pm \sqrt{48 (- (x - 1)) (x - 1)}}{2 \cdot 4}$

Step 5.1.6.3.4

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$y = \frac{40 \pm \sqrt{48 (- 1 (x - 1)) (x - 1)}}{2 \cdot 4}$

Step 5.1.6.3.5

Raise $x - 1$ to the power of $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 ((x - 1) (x - 1)))}}{2 \cdot 4}$

Step 5.1.6.3.6

Raise $x - 1$ to the power of $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 ((x - 1) (x - 1)))}}{2 \cdot 4}$

Step 5.1.6.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 {(x - 1)}^{1 + 1})}}{2 \cdot 4}$

Step 5.1.6.3.8

Add $1$ and $1$ .

$y = \frac{40 \pm \sqrt{48 \cdot (- 1 {(x - 1)}^{2})}}{2 \cdot 4}$

Step 5.1.6.3.9

Multiply $48$ by $- 1$ .

$y = \frac{40 \pm \sqrt{- 48 {(x - 1)}^{2}}}{2 \cdot 4}$

Step 5.1.7

Rewrite $- 48 {(x - 1)}^{2}$ as ${(4 (x - 1))}^{2} \cdot - 3$ .

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Step 5.1.7.1

Factor $16$ out of $- 48$ .

$y = \frac{40 \pm \sqrt{16 (- 3) {(x - 1)}^{2}}}{2 \cdot 4}$

Step 5.1.7.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{40 \pm \sqrt{4^{2} \cdot (- 3 {(x - 1)}^{2})}}{2 \cdot 4}$

Step 5.1.7.3

Move $- 3$ .

$y = \frac{40 \pm \sqrt{4^{2} {(x - 1)}^{2} \cdot - 3}}{2 \cdot 4}$

Step 5.1.7.4

Rewrite $4^{2} {(x - 1)}^{2}$ as ${(4 (x - 1))}^{2}$ .

$y = \frac{40 \pm \sqrt{{(4 (x - 1))}^{2} \cdot - 3}}{2 \cdot 4}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{40 \pm 4 (x - 1) \sqrt{- 3}}{2 \cdot 4}$

Step 5.1.9

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{40 \pm 4 (x - 1) \sqrt{- 1 \cdot 3}}{2 \cdot 4}$

Step 5.1.10

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$y = \frac{40 \pm 4 (x - 1) (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot 4}$

Step 5.1.11

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{40 \pm 4 (x - 1) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 5.1.12

Apply the distributive property.

$y = \frac{40 \pm (4 x + 4 \cdot - 1) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 5.1.13

Multiply $4$ by $- 1$ .

$y = \frac{40 \pm (4 x - 4) (i \cdot \sqrt{3})}{2 \cdot 4}$

Step 5.1.14

Apply the distributive property.

$y = \frac{40 \pm (4 x i \sqrt{3} - 4 i \sqrt{3})}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$y = \frac{40 \pm (4 x i \sqrt{3} - 4 i \sqrt{3})}{8}$

Step 5.3

Change the $\pm$ to $-$ .

$y = \frac{40 - (4 x i \sqrt{3} - 4 i \sqrt{3})}{8}$

Step 5.4

Cancel the common factor of $40 - (4 x i \sqrt{3} - 4 i \sqrt{3})$ and $8$ .

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Step 5.4.1

Rewrite $40$ as $- 1 (- 40)$ .

$y = \frac{- 1 \cdot - 40 - (4 x i \sqrt{3} - 4 i \sqrt{3})}{8}$

Step 5.4.2

Factor $- 1$ out of $- 1 (- 40) - (4 x i \sqrt{3} - 4 i \sqrt{3})$ .

$y = \frac{- 1 (- 40 + 4 x i \sqrt{3} - 4 i \sqrt{3})}{8}$

Step 5.4.3

Factor $4$ out of $- 1 (- 40 + 4 x i \sqrt{3} - 4 i \sqrt{3})$ .

$y = \frac{4 (- 1 (- 10 + x i \sqrt{3} - i \sqrt{3}))}{8}$

Step 5.4.4

Cancel the common factors.

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Step 5.4.4.1

Factor $4$ out of $8$ .

$y = \frac{4 (- 1 (- 10 + x i \sqrt{3} - i \sqrt{3}))}{4 (2)}$

Step 5.4.4.2

Cancel the common factor.

$y = \frac{4 (- 1 (- 10 + x i \sqrt{3} - i \sqrt{3}))}{4 \cdot 2}$

Step 5.4.4.3

Rewrite the expression.

$y = \frac{- 1 (- 10 + x i \sqrt{3} - i \sqrt{3})}{2}$

Step 5.5

Reorder terms.

$y = \frac{- 1 (i x \sqrt{3} - i \sqrt{3} - 10)}{2}$

Step 5.6

Move the negative in front of the fraction.

$y = - \frac{i x \sqrt{3} - i \sqrt{3} - 10}{2}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{i x \sqrt{3} - i \sqrt{3} + 10}{2}$

$y = - \frac{i x \sqrt{3} - i \sqrt{3} - 10}{2}$

Step 7

The given equation $y = \frac{i x \sqrt{3} - i \sqrt{3} + 10}{2}, - \frac{i x \sqrt{3} - i \sqrt{3} - 10}{2}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$