To solve the quadratic equation [tex]\(6x^2 - 12x - 210 = 0\)[/tex], let's go through the steps in detail to find the solutions, i.e., the values of [tex]\(x\)[/tex].
### Step 1: Identify the coefficients
First, we identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].
For the given equation [tex]\(6x^2 - 12x - 210 = 0\)[/tex]:
- [tex]\(a = 6\)[/tex]
- [tex]\(b = -12\)[/tex]
- [tex]\(c = -210\)[/tex]
### Step 2: Compute the discriminant
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-12)^2 - 4 \cdot 6 \cdot (-210) \][/tex]
[tex]\[ \Delta = 144 + 5040 \][/tex]
[tex]\[ \Delta = 5184 \][/tex]
Since the discriminant is positive ([tex]\(\Delta > 0\)[/tex]), the quadratic equation has two distinct real roots.
### Step 3: Find the roots using the quadratic formula
The quadratic formula to find the roots of the equation [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Root 1 ([tex]\(x_1\)[/tex]):
[tex]\[ x_1 = \frac{-(-12) + \sqrt{5184}}{2 \cdot 6} \][/tex]
[tex]\[ x_1 = \frac{12 + 72}{12} \][/tex]
[tex]\[ x_1 = \frac{84}{12} \][/tex]
[tex]\[ x_1 = 7 \][/tex]
Root 2 ([tex]\(x_2\)[/tex]):
[tex]\[ x_2 = \frac{-(-12) - \sqrt{5184}}{2 \cdot 6} \][/tex]
[tex]\[ x_2 = \frac{12 - 72}{12} \][/tex]
[tex]\[ x_2 = \frac{-60}{12} \][/tex]
[tex]\[ x_2 = -5 \][/tex]
### Step 4: Order the solutions
The solutions should be listed from least to greatest.
- Lesser [tex]\(x\)[/tex] is [tex]\(-5\)[/tex]
- Greater [tex]\(x\)[/tex] is [tex]\(7\)[/tex]
### Answer:
- Lesser [tex]\(x = -5\)[/tex]
- Greater [tex]\(x = 7\)[/tex]
Thus, the solutions to the equation [tex]\(6x^2 - 12x - 210 = 0\)[/tex] are [tex]\(x = -5\)[/tex] and [tex]\(x = 7\)[/tex], in ascending order.
