Algebra Examples

$x^{2} + 3 x + c = \frac{7}{4} + c$

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

Subtract $\frac{7}{4}$ from both sides of the equation.

$x^{2} + 3 x + c - \frac{7}{4} = c$

Step 1.1.2

Subtract $c$ from both sides of the equation.

$x^{2} + 3 x + c - \frac{7}{4} - c = 0$

Step 1.2

Combine the opposite terms in $x^{2} + 3 x + c - \frac{7}{4} - c$ .

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Step 1.2.1

Subtract $c$ from $c$ .

$x^{2} + 3 x + 0 - \frac{7}{4} = 0$

Step 1.2.2

Add $x^{2} + 3 x$ and $0$ .

$x^{2} + 3 x - \frac{7}{4} = 0$

Step 2

Multiply through by the least common denominator $4$ , then simplify.

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Step 2.1

Apply the distributive property.

$4 x^{2} + 4 (3 x) + 4 (- \frac{7}{4}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $3$ by $4$ .

$4 x^{2} + 12 x + 4 (- \frac{7}{4}) = 0$

Step 2.2.2

Cancel the common factor of $4$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$4 x^{2} + 12 x + 4 (\frac{- 7}{4}) = 0$

Step 2.2.2.2

Cancel the common factor.

$4 x^{2} + 12 x + 4 (\frac{- 7}{4}) = 0$

Step 2.2.2.3

Rewrite the expression.

$4 x^{2} + 12 x - 7 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = 12$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (4 \cdot - 7)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 12 \pm \sqrt{144 - 16 \cdot - 7}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{- 12 \pm \sqrt{144 + 112}}{2 \cdot 4}$

Step 5.1.3

Add $144$ and $112$ .

$x = \frac{- 12 \pm \sqrt{256}}{2 \cdot 4}$

Step 5.1.4

Rewrite $256$ as $16^{2}$ .

$x = \frac{- 12 \pm \sqrt{16^{2}}}{2 \cdot 4}$

Step 5.1.5

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

$x = \frac{- 12 \pm 16}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{- 12 \pm 16}{8}$

Step 5.3

Simplify $\frac{- 12 \pm 16}{8}$ .

$x = \frac{- 3 \pm 4}{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 12 \pm \sqrt{144 - 16 \cdot - 7}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{- 12 \pm \sqrt{144 + 112}}{2 \cdot 4}$

Step 6.1.3

Add $144$ and $112$ .

$x = \frac{- 12 \pm \sqrt{256}}{2 \cdot 4}$

Step 6.1.4

Rewrite $256$ as $16^{2}$ .

$x = \frac{- 12 \pm \sqrt{16^{2}}}{2 \cdot 4}$

Step 6.1.5

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

$x = \frac{- 12 \pm 16}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{- 12 \pm 16}{8}$

Step 6.3

Simplify $\frac{- 12 \pm 16}{8}$ .

$x = \frac{- 3 \pm 4}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{- 3 + 4}{2}$

Step 6.5

Add $- 3$ and $4$ .

$x = \frac{1}{2}$

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 $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 4 \cdot - 7}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 12 \pm \sqrt{144 - 16 \cdot - 7}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{- 12 \pm \sqrt{144 + 112}}{2 \cdot 4}$

Step 7.1.3

Add $144$ and $112$ .

$x = \frac{- 12 \pm \sqrt{256}}{2 \cdot 4}$

Step 7.1.4

Rewrite $256$ as $16^{2}$ .

$x = \frac{- 12 \pm \sqrt{16^{2}}}{2 \cdot 4}$

Step 7.1.5

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

$x = \frac{- 12 \pm 16}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{- 12 \pm 16}{8}$

Step 7.3

Simplify $\frac{- 12 \pm 16}{8}$ .

$x = \frac{- 3 \pm 4}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{- 3 - 4}{2}$

Step 7.5

Subtract $4$ from $- 3$ .

$x = \frac{- 7}{2}$

Step 7.6

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{1}{2}, - \frac{7}{2}$

Step 9

To rewrite as a function of $c$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $c$ is on the other side.

$f (c) = \frac{1}{2}, - \frac{7}{2}$