Basic Math Examples

$3 a^{2} + 10 a b + 7 b^{2} = 0$

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 $a = 7$ , $b = 10 a$ , and $c = 3 a^{2}$ into the quadratic formula and solve for $b$ .

$\frac{- 10 a \pm \sqrt{{(10 a)}^{2} - 4 \cdot (7 \cdot (3 a^{2}))}}{2 \cdot 7}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add parentheses.

$b = \frac{- 10 a \pm \sqrt{{(10 a)}^{2} - 4 \cdot (7 \cdot (3 a^{2}))}}{2 \cdot 7}$

Step 3.1.2

Let $u = 7 \cdot (3 a^{2})$ . Substitute $u$ for all occurrences of $7 \cdot (3 a^{2})$ .

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Step 3.1.2.1

Apply the product rule to $10 a$ .

$b = \frac{- 10 a \pm \sqrt{10^{2} a^{2} - 4 \cdot u}}{2 \cdot 7}$

Step 3.1.2.2

Raise $10$ to the power of $2$ .

$b = \frac{- 10 a \pm \sqrt{100 a^{2} - 4 u}}{2 \cdot 7}$

Step 3.1.3

Factor $4$ out of $100 a^{2} - 4 u$ .

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Step 3.1.3.1

Factor $4$ out of $100 a^{2}$ .

$b = \frac{- 10 a \pm \sqrt{4 (25 a^{2}) - 4 u}}{2 \cdot 7}$

Step 3.1.3.2

Factor $4$ out of $- 4 u$ .

$b = \frac{- 10 a \pm \sqrt{4 (25 a^{2}) + 4 (- u)}}{2 \cdot 7}$

Step 3.1.3.3

Factor $4$ out of $4 (25 a^{2}) + 4 (- u)$ .

$b = \frac{- 10 a \pm \sqrt{4 (25 a^{2} - u)}}{2 \cdot 7}$

Step 3.1.4

Replace all occurrences of $u$ with $7 \cdot (3 a^{2})$ .

$b = \frac{- 10 a \pm \sqrt{4 (25 a^{2} - (7 \cdot (3 a^{2})))}}{2 \cdot 7}$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

Multiply $3$ by $7$ .

$b = \frac{- 10 a \pm \sqrt{4 (25 a^{2} - (21 a^{2}))}}{2 \cdot 7}$

Step 3.1.5.1.2

Multiply $21$ by $- 1$ .

$b = \frac{- 10 a \pm \sqrt{4 (25 a^{2} - 21 a^{2})}}{2 \cdot 7}$

Step 3.1.5.2

Subtract $21 a^{2}$ from $25 a^{2}$ .

$b = \frac{- 10 a \pm \sqrt{4 (4 a^{2})}}{2 \cdot 7}$

$b = \frac{- 10 a \pm \sqrt{4 \cdot (4 a^{2})}}{2 \cdot 7}$

Step 3.1.6

Multiply $4$ by $4$ .

$b = \frac{- 10 a \pm \sqrt{16 a^{2}}}{2 \cdot 7}$

Step 3.1.7

Rewrite $16 a^{2}$ as ${(4 a)}^{2}$ .

$b = \frac{- 10 a \pm \sqrt{{(4 a)}^{2}}}{2 \cdot 7}$

Step 3.1.8

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

$b = \frac{- 10 a \pm 4 a}{2 \cdot 7}$

Step 3.2

Multiply $2$ by $7$ .

$b = \frac{- 10 a \pm 4 a}{14}$

Step 3.3

Simplify $\frac{- 10 a \pm 4 a}{14}$ .

$b = \frac{- 5 a \pm 2 a}{7}$

Step 4

The final answer is the combination of both solutions.

$b = - \frac{3 a}{7}$

$b = - a$