Algebra Examples

$x^{2} - 9 x + 9 < y$

Step 1

Subtract $y$ from both sides of the inequality.

$x^{2} - 9 x + 9 - y < 0$

Step 2

Convert the inequality to an equation.

$x^{2} - 9 x + 9 - y = 0$

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 $a = 1$ , $b = - 9$ , and $c = 9 - y$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot (9 - y))}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot (9 - y)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot (9 - y)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 9 - 4 (- y)}}{2 \cdot 1}$

Step 5.1.4

Multiply $- 4$ by $9$ .

$x = \frac{9 \pm \sqrt{81 - 36 - 4 (- y)}}{2 \cdot 1}$

Step 5.1.5

Multiply $- 1$ by $- 4$ .

$x = \frac{9 \pm \sqrt{81 - 36 + 4 y}}{2 \cdot 1}$

Step 5.1.6

Subtract $36$ from $81$ .

$x = \frac{9 \pm \sqrt{45 + 4 y}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{45 + 4 y}}{2}$

Step 6

Consolidate the solutions.

$x = \frac{9 + \sqrt{45 + 4 y}}{2}$

$x = \frac{9 - \sqrt{45 + 4 y}}{2}$