Solved on Dec 05, 2023

Solve $7 x^{2} + 1 = -4 x$ over complex numbers. Solution set is $x = \frac{-1 \pm \sqrt{1 - 28}}{14}$ .

STEP 1

Assumptions
1. We are solving the quadratic equation $7x^{2} + 1 = -4x$ .
2. The solutions are to be found over the set of complex numbers.

STEP 2

To solve the quadratic equation, we first need to bring all terms to one side of the equation to set it equal to zero.
$7x^{2} + 4x + 1 = 0$

STEP 3

Now we will attempt to factor the quadratic, but if that is not possible, we will use the quadratic formula to find the roots. The quadratic formula is given by:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$ .

STEP 4

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $7x^{2} + 4x + 1 = 0$ .
$a = 7, b = 4, c = 1$

STEP 5

Plug the coefficients $a$ , $b$ , and $c$ into the quadratic formula.
$x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 7 \cdot 1}}{2 \cdot 7}$

STEP 6

Simplify the expression under the square root (the discriminant).
$\Delta = 4^2 - 4 \cdot 7 \cdot 1 = 16 - 28 = -12$

STEP 7

Since the discriminant $\Delta$ is negative, we will have two complex roots. We can express the square root of a negative number using the imaginary unit $i$ where $i^2 = -1$ .
$\sqrt{-12} = \sqrt{12} \cdot i = 2\sqrt{3} \cdot i$

STEP 8

Substitute the value of the discriminant back into the quadratic formula.
$x = \frac{-4 \pm 2\sqrt{3} \cdot i}{14}$

STEP 9

Simplify the fraction by dividing both the numerator and the denominator by 2.
$x = \frac{-2 \pm \sqrt{3} \cdot i}{7}$

STEP 10

Now we have the two solutions for the quadratic equation, which are complex numbers.
$x_1 = \frac{-2 + \sqrt{3} \cdot i}{7}$ $x_2 = \frac{-2 - \sqrt{3} \cdot i}{7}$
The solution set is $\left\{ \frac{-2 + \sqrt{3} \cdot i}{7}, \frac{-2 - \sqrt{3} \cdot i}{7} \right\}$ .

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Solve $7 x^{2} + 1 = -4 x$ over complex numbers. Solution set is $x = \frac{-1 \pm \sqrt{1 - 28}}{14}$ .

STEP 1

Assumptions
1. We are solving the quadratic equation $7x^{2} + 1 = -4x$ .
2. The solutions are to be found over the set of complex numbers.

STEP 2

To solve the quadratic equation, we first need to bring all terms to one side of the equation to set it equal to zero.
$7x^{2} + 4x + 1 = 0$

STEP 3

Now we will attempt to factor the quadratic, but if that is not possible, we will use the quadratic formula to find the roots. The quadratic formula is given by:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$ .

STEP 4

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $7x^{2} + 4x + 1 = 0$ .
$a = 7, b = 4, c = 1$

STEP 5

Plug the coefficients $a$ , $b$ , and $c$ into the quadratic formula.
$x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 7 \cdot 1}}{2 \cdot 7}$

STEP 6

Simplify the expression under the square root (the discriminant).
$\Delta = 4^2 - 4 \cdot 7 \cdot 1 = 16 - 28 = -12$

STEP 7

Since the discriminant $\Delta$ is negative, we will have two complex roots. We can express the square root of a negative number using the imaginary unit $i$ where $i^2 = -1$ .
$\sqrt{-12} = \sqrt{12} \cdot i = 2\sqrt{3} \cdot i$

STEP 8

Substitute the value of the discriminant back into the quadratic formula.
$x = \frac{-4 \pm 2\sqrt{3} \cdot i}{14}$

STEP 9

Simplify the fraction by dividing both the numerator and the denominator by 2.
$x = \frac{-2 \pm \sqrt{3} \cdot i}{7}$

STEP 10

Now we have the two solutions for the quadratic equation, which are complex numbers.
$x_1 = \frac{-2 + \sqrt{3} \cdot i}{7}$ $x_2 = \frac{-2 - \sqrt{3} \cdot i}{7}$
The solution set is $\left\{ \frac{-2 + \sqrt{3} \cdot i}{7}, \frac{-2 - \sqrt{3} \cdot i}{7} \right\}$ .

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Solve 7x2+1=−4x7 x^{2} + 1 = -4 x7x2+1=−4x over complex numbers. Solution set is x=−1±1−2814x = \frac{-1 \pm \sqrt{1 - 28}}{14}x=14−1±1−28​​.

STEP 1

Assumptions1. We are solving the quadratic equation 7x2+1=−4x7x^{2} + 1 = -4x7x2+1=−4x.2. The solutions are to be found over the set of complex numbers.

STEP 2

To solve the quadratic equation, we first need to bring all terms to one side of the equation to set it equal to zero.7x2+4x+1=07x^{2} + 4x + 1 = 07x2+4x+1=0

STEP 3

STEP 4

Identify the coefficients aaa, bbb, and ccc from the quadratic equation 7x2+4x+1=07x^{2} + 4x + 1 = 07x2+4x+1=0.a=7,b=4,c=1a = 7, b = 4, c = 1a=7,b=4,c=1

STEP 5

Plug the coefficients aaa, bbb, and ccc into the quadratic formula.x=−4±42−4⋅7⋅12⋅7x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 7 \cdot 1}}{2 \cdot 7}x=2⋅7−4±42−4⋅7⋅1​​

STEP 6

Simplify the expression under the square root (the discriminant).Δ=42−4⋅7⋅1=16−28=−12\Delta = 4^2 - 4 \cdot 7 \cdot 1 = 16 - 28 = -12Δ=42−4⋅7⋅1=16−28=−12

STEP 7

Since the discriminant Δ\DeltaΔ is negative, we will have two complex roots. We can express the square root of a negative number using the imaginary unit iii where i2=−1i^2 = -1i2=−1.−12=12⋅i=23⋅i\sqrt{-12} = \sqrt{12} \cdot i = 2\sqrt{3} \cdot i−12​=12​⋅i=23​⋅i

STEP 8

Substitute the value of the discriminant back into the quadratic formula.x=−4±23⋅i14x = \frac{-4 \pm 2\sqrt{3} \cdot i}{14}x=14−4±23​⋅i​

STEP 9

Simplify the fraction by dividing both the numerator and the denominator by 2.x=−2±3⋅i7x = \frac{-2 \pm \sqrt{3} \cdot i}{7}x=7−2±3​⋅i​

STEP 10

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Solve $7 x^{2} + 1 = -4 x$ over complex numbers. Solution set is $x = \frac{-1 \pm \sqrt{1 - 28}}{14}$ .

Assumptions
1. We are solving the quadratic equation $7x^{2} + 1 = -4x$ .
2. The solutions are to be found over the set of complex numbers.

To solve the quadratic equation, we first need to bring all terms to one side of the equation to set it equal to zero.
$7x^{2} + 4x + 1 = 0$

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $7x^{2} + 4x + 1 = 0$ .
$a = 7, b = 4, c = 1$

Plug the coefficients $a$ , $b$ , and $c$ into the quadratic formula.
$x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 7 \cdot 1}}{2 \cdot 7}$

Simplify the expression under the square root (the discriminant).
$\Delta = 4^2 - 4 \cdot 7 \cdot 1 = 16 - 28 = -12$

Since the discriminant $\Delta$ is negative, we will have two complex roots. We can express the square root of a negative number using the imaginary unit $i$ where $i^2 = -1$ .
$\sqrt{-12} = \sqrt{12} \cdot i = 2\sqrt{3} \cdot i$

Substitute the value of the discriminant back into the quadratic formula.
$x = \frac{-4 \pm 2\sqrt{3} \cdot i}{14}$

Simplify the fraction by dividing both the numerator and the denominator by 2.
$x = \frac{-2 \pm \sqrt{3} \cdot i}{7}$