What is -2x^2+6x<=-21 ?

The solution to -2x^2+6x = (sqrt(51)+3)/2

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-2x^2+6x<=-21

- 2 x^{2} + 6 x \leq - 21

x \leq \frac{- 51 + 3}{2} or x \geq \frac{51 + 3}{2}

Interval Notation

(- \infty, \frac{- 51 + 3}{2}] \cup [\frac{51 + 3}{2}, \infty)

Decimal Notation

x \leq - 2.07071 \dots or x \geq 5.07071 \dots

Solution steps

- 2 x^{2} + 6 x \leq - 21

Rewrite in standard form

- 2 x^{2} + 6 x + 21 \leq 0

Complete the square

- 2 x^{2} + 6 x + 21 : - 2 (x - \frac{3}{2})^{2} + \frac{51}{2}

- 2 (x - \frac{3}{2})^{2} + \frac{51}{2} \leq 0

Move

\frac{51}{2}

to the right side

- 2 (x - \frac{3}{2})^{2} \leq - \frac{51}{2}

Multiply both sides by

- 1

2 (x - \frac{3}{2})^{2} \geq \frac{51}{2}

Divide both sides by

2

(x - \frac{3}{2})^{2} \geq \frac{51}{4}

For

u^{n} \geq a

, if

n

is even then

x - \frac{3}{2} \leq - \frac{51}{4} or x - \frac{3}{2} \geq \frac{51}{4}

x - \frac{3}{2} \leq - \frac{51}{4} : x \leq \frac{- 51 + 3}{2}

x - \frac{3}{2} \geq \frac{51}{4} : x \geq \frac{51 + 3}{2}

Combine the intervals

x \leq \frac{- 51 + 3}{2} or x \geq \frac{51 + 3}{2}

\frac{4 x - 5}{5} = 3

x^{2} - 100 x = 0

\frac{4 x + 6}{5} \geq \frac{8 x - 8}{6}

6 = \frac{3 x - 1}{2} \cdot 2.4

8 1^{\frac{- 1}{2}}

What is -2x^2+6x<=-21 ?
The solution to -2x^2+6x<=-21 is x<= (-sqrt(51)+3)/2 \lor x>= (sqrt(51)+3)/2