Precalculus Examples

$2 x^{2} - 7 x = 5$

Divide each term in $2 x^{2} - 7 x = 5$ by $2$ and simplify.

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Divide each term in $2 x^{2} - 7 x = 5$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{- 7 x}{2} = \frac{5}{2}$

Simplify the left side.

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Simplify each term.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{- 7 x}{2} = \frac{5}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{- 7 x}{2} = \frac{5}{2}$

Move the negative in front of the fraction.

$x^{2} - \frac{7 x}{2} = \frac{5}{2}$

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{7}{4})}^{2}$

Add the term to each side of the equation.

$x^{2} - \frac{7 x}{2} + {(- \frac{7}{4})}^{2} = \frac{5}{2} + {(- \frac{7}{4})}^{2}$

Simplify the equation.

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Simplify the left side.

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Simplify each term.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{7}{4}$ .

$x^{2} - \frac{7 x}{2} + {(- 1)}^{2} {(\frac{7}{4})}^{2} = \frac{5}{2} + {(- \frac{7}{4})}^{2}$

Apply the product rule to $\frac{7}{4}$ .

$x^{2} - \frac{7 x}{2} + {(- 1)}^{2} \frac{7^{2}}{4^{2}} = \frac{5}{2} + {(- \frac{7}{4})}^{2}$

Raise $- 1$ to the power of $2$ .

$x^{2} - \frac{7 x}{2} + 1 \frac{7^{2}}{4^{2}} = \frac{5}{2} + {(- \frac{7}{4})}^{2}$

Multiply $\frac{7^{2}}{4^{2}}$ by $1$ .

$x^{2} - \frac{7 x}{2} + \frac{7^{2}}{4^{2}} = \frac{5}{2} + {(- \frac{7}{4})}^{2}$

Raise $7$ to the power of $2$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{4^{2}} = \frac{5}{2} + {(- \frac{7}{4})}^{2}$

Raise $4$ to the power of $2$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5}{2} + {(- \frac{7}{4})}^{2}$

Simplify the right side.

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Simplify $\frac{5}{2} + {(- \frac{7}{4})}^{2}$ .

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Simplify each term.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{7}{4}$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5}{2} + {(- 1)}^{2} {(\frac{7}{4})}^{2}$

Apply the product rule to $\frac{7}{4}$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5}{2} + {(- 1)}^{2} \frac{7^{2}}{4^{2}}$

Raise $- 1$ to the power of $2$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5}{2} + 1 \frac{7^{2}}{4^{2}}$

Multiply $\frac{7^{2}}{4^{2}}$ by $1$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5}{2} + \frac{7^{2}}{4^{2}}$

Raise $7$ to the power of $2$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5}{2} + \frac{49}{4^{2}}$

Raise $4$ to the power of $2$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5}{2} + \frac{49}{16}$

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5}{2} \cdot \frac{8}{8} + \frac{49}{16}$

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{5}{2}$ by $\frac{8}{8}$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5 \cdot 8}{2 \cdot 8} + \frac{49}{16}$

Multiply $2$ by $8$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5 \cdot 8}{16} + \frac{49}{16}$

Combine the numerators over the common denominator.

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{5 \cdot 8 + 49}{16}$

Simplify the numerator.

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Multiply $5$ by $8$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{40 + 49}{16}$

Add $40$ and $49$ .

$x^{2} - \frac{7 x}{2} + \frac{49}{16} = \frac{89}{16}$

Factor the perfect trinomial square into ${(x - \frac{7}{4})}^{2}$ .

${(x - \frac{7}{4})}^{2} = \frac{89}{16}$

Solve the equation for $x$ .

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Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x - \frac{7}{4} = \pm \sqrt{\frac{89}{16}}$

Simplify $\pm \sqrt{\frac{89}{16}}$ .

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Rewrite $\sqrt{\frac{89}{16}}$ as $\frac{\sqrt{89}}{\sqrt{16}}$ .

$x - \frac{7}{4} = \pm \frac{\sqrt{89}}{\sqrt{16}}$

Simplify the denominator.

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Rewrite $16$ as $4^{2}$ .

$x - \frac{7}{4} = \pm \frac{\sqrt{89}}{\sqrt{4^{2}}}$

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

$x - \frac{7}{4} = \pm \frac{\sqrt{89}}{4}$

Add $\frac{7}{4}$ to both sides of the equation.

$x = \pm \frac{\sqrt{89}}{4} + \frac{7}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{89}}{4} + \frac{7}{4}$

Decimal Form:

$x = 4.10849528 \dots, - 0.60849528 \dots$