Algebra Examples

$11 x^{2} = - 5 x^{2} - 1$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

Add $5 x^{2}$ to both sides of the equation.

$11 x^{2} + 5 x^{2} = - 1$

Step 1.1.2

Add $1$ to both sides of the equation.

$11 x^{2} + 5 x^{2} + 1 = 0$

Step 1.2

Add $11 x^{2}$ and $5 x^{2}$ .

$16 x^{2} + 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 16$ , $b = 0$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (16 \cdot 1)}}{2 \cdot 16}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 16 \cdot 1}}{2 \cdot 16}$

Step 4.1.2

Multiply $- 4 \cdot 16 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{0 \pm \sqrt{0 - 64 \cdot 1}}{2 \cdot 16}$

Step 4.1.2.2

Multiply $- 64$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 64}}{2 \cdot 16}$

Step 4.1.3

Subtract $64$ from $0$ .

$x = \frac{0 \pm \sqrt{- 64}}{2 \cdot 16}$

Step 4.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 16}$

Step 4.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 16}$

Step 4.1.6

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

$x = \frac{0 \pm i \cdot \sqrt{64}}{2 \cdot 16}$

Step 4.1.7

Rewrite $64$ as $8^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 16}$

Step 4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{0 \pm i \cdot 8}{2 \cdot 16}$

Step 4.1.9

Move $8$ to the left of $i$ .

$x = \frac{0 \pm 8 i}{2 \cdot 16}$

Step 4.2

Multiply $2$ by $16$ .

$x = \frac{0 \pm 8 i}{32}$

Step 4.3

Simplify $\frac{0 \pm 8 i}{32}$ .

$x = \frac{\pm i}{4}$