Question

X+8<3x^2-9x через дискриминант

Answer 1

Смотри..................

Answer 2

To solve the inequality $x + 8 < 3x^2 - 9x$ using the discriminant, we first need to rewrite the inequality in the form $ax^2 + bx + c < 0$. The discriminant is then used to determine the intervals where the inequality is satisfied.

Step 1: Rewrite the inequality in the form $ax^2 + bx + c < 0$:

$3x^2 - 9x - (x + 8) < 0$

Step 2: Simplify the expression:

$3x^2 - 9x - x - 8 < 0$

$3x^2 - 10x - 8 < 0$

Now, the inequality is in the form $ax^2 + bx + c < 0$, where $a = 3$, $b = -10$, and $c = -8$.

Step 3: Use the discriminant to find the critical points:

The discriminant ($\Delta$) is given by: $\Delta = b^2 - 4ac$

$\Delta = (-10)^2 - 4 * 3 * (-8)$

$\Delta = 100 + 96$

$\Delta = 196$

Step 4: Find the critical points using the quadratic formula:

The critical points are given by $x = \frac{-b \pm \sqrt{\Delta}}{2a}$

$x = \frac{-(-10) \pm \sqrt{196}}{2 * 3}$

$x = \frac{10 \pm 14}{6}$

So, the critical points are $x = \frac{24}{6} = 4$ and $x = \frac{-4}{6} = -\frac{2}{3}$.

Step 5: Determine the intervals where the inequality is satisfied:

Now we need to determine in which intervals the quadratic expression $3x^2 - 10x - 8$ is less than zero.

We have three intervals to consider:

a) $x < -\frac{2}{3}$ b) $-\frac{2}{3} < x < 4$ c) $x > 4$

Step 6: Test a point in each interval to check if it satisfies the inequality:

a) Let's choose $x = -1$ (a value less than $-\frac{2}{3}$):

$3(-1)^2 - 10(-1) - 8 = 3 + 10 - 8 = 5$

Since $5$ is positive, the inequality is not satisfied in this interval.

b) Let's choose $x = 0$ (a value between $-\frac{2}{3}$ and $4$):

$3(0)^2 - 10(0) - 8 = -8$

Since $-8$ is negative, the inequality is satisfied in this interval.

c) Let's choose $x = 5$ (a value greater than $4$):

$3(5)^2 - 10(5) - 8 = 75 - 50 - 8 = 17$

Since $17$ is positive, the inequality is not satisfied in this interval.

Step 7: Final conclusion:

The inequality $x + 8 < 3x^2 - 9x$ is satisfied in the interval $-\frac{2}{3} < x < 4$.