Basic Math Examples

$2 a^{2} + a b + b^{2} = 4$

Step 1

Subtract $4$ from both sides of the equation.

$2 a^{2} + a b + b^{2} - 4 = 0$

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 $a = 1$ , $b = a$ , and $c = 2 a^{2} - 4$ into the quadratic formula and solve for $b$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 4$ by $1$ .

$b = \frac{- a \pm \sqrt{a^{2} - 4 \cdot (2 a^{2} - 4)}}{2 \cdot 1}$

Step 4.1.2

Apply the distributive property.

$b = \frac{- a \pm \sqrt{a^{2} - 4 (2 a^{2}) - 4 \cdot - 4}}{2 \cdot 1}$

Step 4.1.3

Multiply $2$ by $- 4$ .

$b = \frac{- a \pm \sqrt{a^{2} - 8 a^{2} - 4 \cdot - 4}}{2 \cdot 1}$

Step 4.1.4

Multiply $- 4$ by $- 4$ .

$b = \frac{- a \pm \sqrt{a^{2} - 8 a^{2} + 16}}{2 \cdot 1}$

Step 4.1.5

Subtract $8 a^{2}$ from $a^{2}$ .

$b = \frac{- a \pm \sqrt{- 7 a^{2} + 16}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$b = \frac{- a \pm \sqrt{- 7 a^{2} + 16}}{2}$

Step 5

The final answer is the combination of both solutions.

$b = - \frac{a - \sqrt{- 7 a^{2} + 16}}{2}$

$b = - \frac{a + \sqrt{- 7 a^{2} + 16}}{2}$