Convert to Interval Notation 3x^2+25<10x

$3 x^{2} + 25 < 10 x$

Step 1

Subtract $10 x$ from both sides of the inequality.

$3 x^{2} + 25 - 10 x < 0$

Step 2

Convert the inequality to an equation.

$3 x^{2} + 25 - 10 x = 0$

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 $a = 3$ , $b = - 10$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (3 \cdot 25)}}{2 \cdot 3}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot 25}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 25$ .

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12 \cdot 25}}{2 \cdot 3}$

Multiply $- 12$ by $25$ .

$x = \frac{10 \pm \sqrt{100 - 300}}{2 \cdot 3}$

Subtract $300$ from $100$ .

$x = \frac{10 \pm \sqrt{- 200}}{2 \cdot 3}$

Rewrite $- 200$ as $- 1 (200)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 200}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (200)}$ as $\sqrt{- 1} \cdot \sqrt{200}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{200}}{2 \cdot 3}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{200}}{2 \cdot 3}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$x = \frac{10 \pm i \cdot \sqrt{100 (2)}}{2 \cdot 3}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{10^{2} \cdot 2}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (10 \sqrt{2})}{2 \cdot 3}$

Move $10$ to the left of $i$ .

$x = \frac{10 \pm 10 i \sqrt{2}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{10 \pm 10 i \sqrt{2}}{6}$

Simplify $\frac{10 \pm 10 i \sqrt{2}}{6}$ .

$x = \frac{5 \pm 5 i \sqrt{2}}{3}$

Step 6

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot 25}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 25$ .

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12 \cdot 25}}{2 \cdot 3}$

Multiply $- 12$ by $25$ .

$x = \frac{10 \pm \sqrt{100 - 300}}{2 \cdot 3}$

Subtract $300$ from $100$ .

$x = \frac{10 \pm \sqrt{- 200}}{2 \cdot 3}$

Rewrite $- 200$ as $- 1 (200)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 200}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (200)}$ as $\sqrt{- 1} \cdot \sqrt{200}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{200}}{2 \cdot 3}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{200}}{2 \cdot 3}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$x = \frac{10 \pm i \cdot \sqrt{100 (2)}}{2 \cdot 3}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{10^{2} \cdot 2}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (10 \sqrt{2})}{2 \cdot 3}$

Move $10$ to the left of $i$ .

$x = \frac{10 \pm 10 i \sqrt{2}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{10 \pm 10 i \sqrt{2}}{6}$

Simplify $\frac{10 \pm 10 i \sqrt{2}}{6}$ .

$x = \frac{5 \pm 5 i \sqrt{2}}{3}$

Change the $\pm$ to $+$ .

$x = \frac{5 + 5 i \sqrt{2}}{3}$

Step 7

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot 25}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 25$ .

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12 \cdot 25}}{2 \cdot 3}$

Multiply $- 12$ by $25$ .

$x = \frac{10 \pm \sqrt{100 - 300}}{2 \cdot 3}$

Subtract $300$ from $100$ .

$x = \frac{10 \pm \sqrt{- 200}}{2 \cdot 3}$

Rewrite $- 200$ as $- 1 (200)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 200}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (200)}$ as $\sqrt{- 1} \cdot \sqrt{200}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{200}}{2 \cdot 3}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{200}}{2 \cdot 3}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$x = \frac{10 \pm i \cdot \sqrt{100 (2)}}{2 \cdot 3}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{10^{2} \cdot 2}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (10 \sqrt{2})}{2 \cdot 3}$

Move $10$ to the left of $i$ .

$x = \frac{10 \pm 10 i \sqrt{2}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{10 \pm 10 i \sqrt{2}}{6}$

Simplify $\frac{10 \pm 10 i \sqrt{2}}{6}$ .

$x = \frac{5 \pm 5 i \sqrt{2}}{3}$

Change the $\pm$ to $-$ .

$x = \frac{5 - 5 i \sqrt{2}}{3}$

Step 8

Identify the leading coefficient.

Move $25$ .

$3 x^{2} - 10 x + 25$

The leading term in a polynomial is the term with the highest degree.

$3 x^{2}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$3$

Step 9

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $3 x^{2} + 25 - 10 x$ is always greater than $0$ .

No solution

Convert to Interval Notation 3x^2+25<10x

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