To solve the quadratic equation [tex]\(3x^2 - 5x - 7 = 0\)[/tex], we need to use the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].
For the given equation [tex]\(3x^2 - 5x - 7 = 0\)[/tex], the coefficients are:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = -7\)[/tex]
### Step 1: Calculate the Discriminant
First, we calculate the discriminant, [tex]\(D\)[/tex], which is given by:
[tex]\[D = b^2 - 4ac\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[D = (-5)^2 - 4(3)(-7)\][/tex]
[tex]\[D = 25 + 84\][/tex]
[tex]\[D = 109\][/tex]
So, the discriminant is 109.
### Step 2: Calculate the Roots
Since the discriminant is positive, we have two distinct real roots. These roots can be found using the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{D}}{2a}\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(D\)[/tex]:
[tex]\[x_1 = \frac{-(-5) + \sqrt{109}}{2(3)}\][/tex]
[tex]\[x_1 = \frac{5 + \sqrt{109}}{6}\][/tex]
And for the second root:
[tex]\[x_2 = \frac{-(-5) - \sqrt{109}}{2(3)}\][/tex]
[tex]\[x_2 = \frac{5 - \sqrt{109}}{6}\][/tex]
### Step 3: Simplify and Round the Roots
Let's find the approximate values of the roots to 3 significant figures:
[tex]\[x_1 \approx 2.573\][/tex]
[tex]\[x_2 \approx -0.907\][/tex]
### Conclusion
The solutions to the quadratic equation [tex]\(3x^2 - 5x - 7 = 0\)[/tex] are:
[tex]\[x_1 \approx 2.573\][/tex]
[tex]\[x_2 \approx -0.907\][/tex]
These are the roots rounded to 3 significant figures.
