The given problem asks to find the range of values for $x$ such that the quadratic expression $2x^2 + 5x - 12$ is positive, and also find the minimum value of this expression.

1. Finding the minimum value of the quadratic expression

A quadratic expression of the form $ax^2 + bx + c$ reaches its minimum (or maximum) at the vertex, where the $x$-coordinate is given by:

$x = -\frac{b}{2a}$

For $2x^2 + 5x - 12$, the coefficients are $a = 2$, $b = 5$, and $c = -12$.

$x_{\text{min}} = -\frac{5}{2(2)} = -\frac{5}{4}$

To find the minimum value, substitute $x = -\frac{5}{4}$ into the quadratic expression:

$f\left(-\frac{5}{4}\right) = 2\left(-\frac{5}{4}\right)^2 + 5\left(-\frac{5}{4}\right) - 12$

$= 2\left(\frac{25}{16}\right) + 5\left(-\frac{5}{4}\right) - 12$

$= \frac{50}{16} - \frac{25}{4} - 12$

$= \frac{50}{16} - \frac{100}{16} - \frac{192}{16}$

$= \frac{50 - 100 - 192}{16} = \frac{-242}{16} = -15.125$

Thus, the minimum value of the expression is $-15.125$.

2. Finding when the expression is positive

To determine when $2x^2 + 5x - 12 > 0$, we first find the roots by solving the equation $2x^2 + 5x - 12 = 0$ using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute $a = 2$, $b = 5$, and $c = -12$:

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

$x = \frac{-5 \pm \sqrt{25 + 96}}{4} = \frac{-5 \pm \sqrt{121}}{4}$

$x = \frac{-5 \pm 11}{4}$

The two roots are:

$x_1 = \frac{-5 + 11}{4} = \frac{6}{4} = 1.5$

$x_2 = \frac{-5 - 11}{4} = \frac{-16}{4} = -4$

The quadratic expression changes signs at the roots. Since the coefficient of $x^2$ is positive, the quadratic expression is positive when $x < -4$ or $x > 1.5$.

Final Answer:

• The expression $2x^2 + 5x - 12$ is positive for $x < -4$ or $x > 1.5$.
• The minimum value of the expression is $-15.125$.

Would you like more details on any part of the solution?

Here are 5 related questions:

1. How do you determine whether a quadratic expression opens upwards or downwards?
2. How can you find the vertex of a quadratic equation using the completing the square method?
3. What does the discriminant of a quadratic equation tell you about its roots?
4. How can you solve quadratic inequalities algebraically?
5. How can you graph a quadratic function to visually identify the range of positive values?

Tip: Always check the sign of the leading coefficient to determine whether a parabola opens upwards (minimum) or downwards (maximum).