Trigonometry Examples

$6 x^{2} - 8 = 4 x^{2} + 7 x$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} + 7 x = 6 x^{2} - 8$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $6 x^{2}$ from both sides of the equation.

$4 x^{2} + 7 x - 6 x^{2} = - 8$

Step 2.2

Subtract $6 x^{2}$ from $4 x^{2}$ .

$- 2 x^{2} + 7 x = - 8$

Step 3

Add $8$ to both sides of the equation.

$- 2 x^{2} + 7 x + 8 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 2$ , $b = 7$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (- 2 \cdot 8)}}{2 \cdot - 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 2 \cdot 8}}{2 \cdot - 2}$

Step 6.1.2

Multiply $- 4 \cdot - 2 \cdot 8$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{49 + 8 \cdot 8}}{2 \cdot - 2}$

Step 6.1.2.2

Multiply $8$ by $8$ .

$x = \frac{- 7 \pm \sqrt{49 + 64}}{2 \cdot - 2}$

Step 6.1.3

Add $49$ and $64$ .

$x = \frac{- 7 \pm \sqrt{113}}{2 \cdot - 2}$

Step 6.2

Multiply $2$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{113}}{- 4}$

Step 6.3

Simplify $\frac{- 7 \pm \sqrt{113}}{- 4}$ .

$x = \frac{7 \pm \sqrt{113}}{4}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 2 \cdot 8}}{2 \cdot - 2}$

Step 7.1.2

Multiply $- 4 \cdot - 2 \cdot 8$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{49 + 8 \cdot 8}}{2 \cdot - 2}$

Step 7.1.2.2

Multiply $8$ by $8$ .

$x = \frac{- 7 \pm \sqrt{49 + 64}}{2 \cdot - 2}$

Step 7.1.3

Add $49$ and $64$ .

$x = \frac{- 7 \pm \sqrt{113}}{2 \cdot - 2}$

Step 7.2

Multiply $2$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{113}}{- 4}$

Step 7.3

Simplify $\frac{- 7 \pm \sqrt{113}}{- 4}$ .

$x = \frac{7 \pm \sqrt{113}}{4}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{113}}{4}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 2 \cdot 8}}{2 \cdot - 2}$

Step 8.1.2

Multiply $- 4 \cdot - 2 \cdot 8$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{49 + 8 \cdot 8}}{2 \cdot - 2}$

Step 8.1.2.2

Multiply $8$ by $8$ .

$x = \frac{- 7 \pm \sqrt{49 + 64}}{2 \cdot - 2}$

Step 8.1.3

Add $49$ and $64$ .

$x = \frac{- 7 \pm \sqrt{113}}{2 \cdot - 2}$

Step 8.2

Multiply $2$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{113}}{- 4}$

Step 8.3

Simplify $\frac{- 7 \pm \sqrt{113}}{- 4}$ .

$x = \frac{7 \pm \sqrt{113}}{4}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{113}}{4}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{113}}{4}, \frac{7 - \sqrt{113}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 + \sqrt{113}}{4}, \frac{7 - \sqrt{113}}{4}$

Decimal Form:

$x = 4.40753645 \dots, - 0.90753645 \dots$