Find the range of values of x for which the expression `12x^(2)+7x-10` is negative.

Answer

To find the range of values of \( x \) for which the expression \( 12x^2 + 7x - 10 \) is negative, we will follow these steps:

### Step 1: Identify the quadratic expression
We have the quadratic expression:
\[ y = 12x^2 + 7x - 10 \]

### Step 2: Determine the nature of the parabola
Since the coefficient of \( x^2 \) (which is 12) is positive, the parabola opens upwards. This means that the expression will be negative between its roots.

### Step 3: Find the roots of the quadratic equation
To find the roots, we set the expression equal to zero:
\[ 12x^2 + 7x - 10 = 0 \]
We will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 12 \), \( b = 7 \), and \( c = -10 \).

### Step 4: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = 7^2 = 49 \]
\[ 4ac = 4 \times 12 \times (-10) = -480 \]
Thus,
\[ b^2 - 4ac = 49 + 480 = 529 \]

### Step 5: Apply the quadratic formula
Now we substitute the values into the quadratic formula:
\[ x = \frac{-7 \pm \sqrt{529}}{2 \times 12} \]
Calculating \( \sqrt{529} \):
\[ \sqrt{529} = 23 \]
So we have:
\[ x = \frac{-7 \pm 23}{24} \]

### Step 6: Calculate the two roots
Calculating the two possible values for \( x \):
1. \( x_1 = \frac{-7 + 23}{24} = \frac{16}{24} = \frac{2}{3} \)
2. \( x_2 = \frac{-7 - 23}{24} = \frac{-30}{24} = -\frac{5}{4} \)

### Step 7: Determine the range of \( x \)
Since the parabola opens upwards, the expression \( 12x^2 + 7x - 10 \) is negative between the roots:
\[ -\frac{5}{4} < x < \frac{2}{3} \]

### Final Answer
The range of values of \( x \) for which the expression \( 12x^2 + 7x - 10 \) is negative is:
\[ \boxed{\left(-\frac{5}{4}, \frac{2}{3}\right)} \]