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

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

Step 1

Add $\frac{5}{4}$ to both sides of the equation.

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

Step 2

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

Apply the distributive property.

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

Simplify.

Multiply $- 3$ by $4$ .

$4 u^{2} - 12 u + 4 (\frac{5}{4}) = 0$

Cancel the common factor of $4$ .

Cancel the common factor.

$4 u^{2} - 12 u + 4 (\frac{5}{4}) = 0$

Rewrite the expression.

$4 u^{2} - 12 u + 5 = 0$

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 $c = 5$ into the quadratic formula and solve for $u$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (4 \cdot 5)}}{2 \cdot 4}$

Step 5

Simplify.

Simplify the numerator.

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

$u = \frac{12 \pm \sqrt{144 - 4 \cdot 4 \cdot 5}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot 5$ .

Multiply $- 4$ by $4$ .

$u = \frac{12 \pm \sqrt{144 - 16 \cdot 5}}{2 \cdot 4}$

Multiply $- 16$ by $5$ .

$u = \frac{12 \pm \sqrt{144 - 80}}{2 \cdot 4}$

Subtract $80$ from $144$ .

$u = \frac{12 \pm \sqrt{64}}{2 \cdot 4}$

Rewrite $64$ as $8^{2}$ .

$u = \frac{12 \pm \sqrt{8^{2}}}{2 \cdot 4}$

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

$u = \frac{12 \pm 8}{2 \cdot 4}$

Multiply $2$ by $4$ .

$u = \frac{12 \pm 8}{8}$

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

$u = \frac{3 \pm 2}{2}$

Step 6

The final answer is the combination of both solutions.

$u = \frac{5}{2}, \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$u = \frac{5}{2}, \frac{1}{2}$

Decimal Form:

$u = 2.5, 0.5$

Mixed Number Form:

$u = 2 \frac{1}{2}, \frac{1}{2}$

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

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