Solve for x x^2-3/4x=5

$x^{2} - \frac{3}{4} x = 5$

Step 1

Simplify each term.

Combine $x$ and $\frac{3}{4}$ .

$x^{2} - \frac{x \cdot 3}{4} = 5$

Move $3$ to the left of $x$ .

$x^{2} - \frac{3 x}{4} = 5$

Step 2

Subtract $5$ from both sides of the equation.

$x^{2} - \frac{3 x}{4} - 5 = 0$

Step 3

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

Apply the distributive property.

$4 x^{2} + 4 (- \frac{3 x}{4}) + 4 \cdot - 5 = 0$

Simplify.

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{3 x}{4}$ into the numerator.

$4 x^{2} + 4 (\frac{- 3 x}{4}) + 4 \cdot - 5 = 0$

Cancel the common factor.

$4 x^{2} + 4 (\frac{- 3 x}{4}) + 4 \cdot - 5 = 0$

Rewrite the expression.

$4 x^{2} - 3 x + 4 \cdot - 5 = 0$

Multiply $4$ by $- 5$ .

$4 x^{2} - 3 x - 20 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 4$ , $b = - 3$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (4 \cdot - 20)}}{2 \cdot 4}$

Step 6

Simplify.

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 20}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 20$ .

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 20}}{2 \cdot 4}$

Multiply $- 16$ by $- 20$ .

$x = \frac{3 \pm \sqrt{9 + 320}}{2 \cdot 4}$

Add $9$ and $320$ .

$x = \frac{3 \pm \sqrt{329}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{329}}{8}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{329}}{8}, \frac{3 - \sqrt{329}}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{329}}{8}, \frac{3 - \sqrt{329}}{8}$

Decimal Form:

$x = 2.64229464 \dots, - 1.89229464 \dots$

Solve for x x^2-3/4x=5

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