Algebra Examples

$x = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}$

Step 1

Rewrite the equation as $\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} = x$ .

$\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} = x$

Step 2

Multiply both sides by $2 a$ .

$\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} (2 a) = x (2 a)$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} (2 a)$ .

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Step 3.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} a = x (2 a)$

Step 3.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.2.1

Factor $2$ out of $2 a$ .

$2 \frac{- b + \sqrt{b^{2} - 4 a c}}{2 (a)} a = x (2 a)$

Step 3.1.1.2.2

Cancel the common factor.

$2 \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} a = x (2 a)$

Step 3.1.1.2.3

Rewrite the expression.

$\frac{- b + \sqrt{b^{2} - 4 a c}}{a} a = x (2 a)$

Step 3.1.1.3

Cancel the common factor of $a$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{- b + \sqrt{b^{2} - 4 a c}}{a} a = x (2 a)$

Step 3.1.1.3.2

Rewrite the expression.

$- b + \sqrt{b^{2} - 4 a c} = x (2 a)$

Step 3.1.1.4

Reorder $b^{2}$ and $- 4 a c$ .

$- b + \sqrt{- 4 a c + b^{2}} = x (2 a)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$- b + \sqrt{- 4 a c + b^{2}} = 2 x a$

Step 4

Solve for $a$ .

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Step 4.1

Add $b$ to both sides of the equation.

$\sqrt{- 4 a c + b^{2}} = 2 x a + b$

Step 4.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{- 4 a c + b^{2}}}^{2} = {(2 x a + b)}^{2}$

Step 4.3

Simplify each side of the equation.

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 4 a c + b^{2}}$ as ${(- 4 a c + b^{2})}^{\frac{1}{2}}$ .

${({(- 4 a c + b^{2})}^{\frac{1}{2}})}^{2} = {(2 x a + b)}^{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Simplify ${({(- 4 a c + b^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1

Multiply the exponents in ${({(- 4 a c + b^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- 4 a c + b^{2})}^{\frac{1}{2} \cdot 2} = {(2 x a + b)}^{2}$

Step 4.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.1.2.1

Cancel the common factor.

${(- 4 a c + b^{2})}^{\frac{1}{2} \cdot 2} = {(2 x a + b)}^{2}$

Step 4.3.2.1.1.2.2

Rewrite the expression.

${(- 4 a c + b^{2})}^{1} = {(2 x a + b)}^{2}$

Step 4.3.2.1.2

Simplify.

$- 4 a c + b^{2} = {(2 x a + b)}^{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify ${(2 x a + b)}^{2}$ .

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Step 4.3.3.1.1

Rewrite ${(2 x a + b)}^{2}$ as $(2 x a + b) (2 x a + b)$ .

$- 4 a c + b^{2} = (2 x a + b) (2 x a + b)$

Step 4.3.3.1.2

Expand $(2 x a + b) (2 x a + b)$ using the FOIL Method.

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Step 4.3.3.1.2.1

Apply the distributive property.

$- 4 a c + b^{2} = 2 x a (2 x a + b) + b (2 x a + b)$

Step 4.3.3.1.2.2

Apply the distributive property.

$- 4 a c + b^{2} = 2 x a (2 x a) + 2 x a b + b (2 x a + b)$

Step 4.3.3.1.2.3

Apply the distributive property.

$- 4 a c + b^{2} = 2 x a (2 x a) + 2 x a b + b (2 x a) + b \cdot b$

Step 4.3.3.1.3

Simplify and combine like terms.

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Step 4.3.3.1.3.1

Simplify each term.

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Step 4.3.3.1.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.3.1.3.1.1.1

Move $x$ .

$- 4 a c + b^{2} = 2 (x \cdot x) a (2 a) + 2 x a b + b (2 x a) + b \cdot b$

Step 4.3.3.1.3.1.1.2

Multiply $x$ by $x$ .

$- 4 a c + b^{2} = 2 x^{2} a (2 a) + 2 x a b + b (2 x a) + b \cdot b$

Step 4.3.3.1.3.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 4.3.3.1.3.1.2.1

Move $a$ .

$- 4 a c + b^{2} = 2 x^{2} (a \cdot a) \cdot 2 + 2 x a b + b (2 x a) + b \cdot b$

Step 4.3.3.1.3.1.2.2

Multiply $a$ by $a$ .

$- 4 a c + b^{2} = 2 x^{2} a^{2} \cdot 2 + 2 x a b + b (2 x a) + b \cdot b$

Step 4.3.3.1.3.1.3

Multiply $2$ by $2$ .

$- 4 a c + b^{2} = 4 x^{2} a^{2} + 2 x a b + b (2 x a) + b \cdot b$

Step 4.3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$- 4 a c + b^{2} = 4 x^{2} a^{2} + 2 x a b + 2 b x a + b \cdot b$

Step 4.3.3.1.3.1.5

Multiply $b$ by $b$ .

$- 4 a c + b^{2} = 4 x^{2} a^{2} + 2 x a b + 2 b x a + b^{2}$

Step 4.3.3.1.3.2

Move $b$ .

$- 4 a c + b^{2} = 4 x^{2} a^{2} + 2 a b x + 2 a b x + b^{2}$

Step 4.3.3.1.3.3

Add $2 a b x$ and $2 a b x$ .

$- 4 a c + b^{2} = 4 x^{2} a^{2} + 4 a b x + b^{2}$

Step 4.4

Solve for $a$ .

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Step 4.4.1

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

$4 x^{2} a^{2} + 4 a b x + b^{2} = - 4 a c + b^{2}$

Step 4.4.2

Add $4 a c$ to both sides of the equation.

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

Step 4.4.3

Move all terms not containing $a$ to the right side of the equation.

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Step 4.4.3.1

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

$4 x^{2} a^{2} + 4 a b x + 4 a c = b^{2} - b^{2}$

Step 4.4.3.2

Subtract $b^{2}$ from $b^{2}$ .

$4 x^{2} a^{2} + 4 a b x + 4 a c = 0$

Step 4.4.4

Factor $4 a$ out of $4 x^{2} a^{2} + 4 a b x + 4 a c$ .

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Step 4.4.4.1

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

$4 a (x^{2} a) + 4 a b x + 4 a c = 0$

Step 4.4.4.2

Factor $4 a$ out of $4 a b x$ .

$4 a (x^{2} a) + 4 a (b x) + 4 a c = 0$

Step 4.4.4.3

Factor $4 a$ out of $4 a c$ .

$4 a (x^{2} a) + 4 a (b x) + 4 a (c) = 0$

Step 4.4.4.4

Factor $4 a$ out of $4 a (x^{2} a) + 4 a (b x)$ .

$4 a (x^{2} a + b x) + 4 a (c) = 0$

Step 4.4.4.5

Factor $4 a$ out of $4 a (x^{2} a + b x) + 4 a (c)$ .

$4 a (x^{2} a + b x + c) = 0$

Step 4.4.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a = 0$

$x^{2} a + b x + c = 0$

Step 4.4.6

Set $a$ equal to $0$ .

$a = 0$

Step 4.4.7

Set $x^{2} a + b x + c$ equal to $0$ and solve for $a$ .

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Step 4.4.7.1

Set $x^{2} a + b x + c$ equal to $0$ .

$x^{2} a + b x + c = 0$

Step 4.4.7.2

Solve $x^{2} a + b x + c = 0$ for $a$ .

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Step 4.4.7.2.1

Move all terms not containing $a$ to the right side of the equation.

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Step 4.4.7.2.1.1

Subtract $b x$ from both sides of the equation.

$x^{2} a + c = - b x$

Step 4.4.7.2.1.2

Subtract $c$ from both sides of the equation.

$x^{2} a = - b x - c$

Step 4.4.7.2.2

Divide each term in $x^{2} a = - b x - c$ by $x^{2}$ and simplify.

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Step 4.4.7.2.2.1

Divide each term in $x^{2} a = - b x - c$ by $x^{2}$ .

$\frac{x^{2} a}{x^{2}} = \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.4.7.2.2.2

Simplify the left side.

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Step 4.4.7.2.2.2.1

Cancel the common factor of $x^{2}$ .

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Step 4.4.7.2.2.2.1.1

Cancel the common factor.

$\frac{x^{2} a}{x^{2}} = \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.4.7.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.4.7.2.2.3

Simplify the right side.

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Step 4.4.7.2.2.3.1

Simplify each term.

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Step 4.4.7.2.2.3.1.1

Cancel the common factor of $x$ and $x^{2}$ .

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Step 4.4.7.2.2.3.1.1.1

Factor $x$ out of $- b x$ .

$a = \frac{x (- b)}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.4.7.2.2.3.1.1.2

Cancel the common factors.

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Step 4.4.7.2.2.3.1.1.2.1

Factor $x$ out of $x^{2}$ .

$a = \frac{x (- b)}{x \cdot x} + \frac{- c}{x^{2}}$

Step 4.4.7.2.2.3.1.1.2.2

Cancel the common factor.

$a = \frac{x (- b)}{x \cdot x} + \frac{- c}{x^{2}}$

Step 4.4.7.2.2.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{x} + \frac{- c}{x^{2}}$

Step 4.4.7.2.2.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{x} + \frac{- c}{x^{2}}$

Step 4.4.7.2.2.3.1.3

Move the negative in front of the fraction.

$a = - \frac{b}{x} - \frac{c}{x^{2}}$

Step 4.4.8

The final solution is all the values that make $4 a (x^{2} a + b x + c) = 0$ true.

$a = 0$

$a = - \frac{b}{x} - \frac{c}{x^{2}}$

$a = 0$

$a = - \frac{b}{x} - \frac{c}{x^{2}}$

$a = 0$

$a = - \frac{b}{x} - \frac{c}{x^{2}}$