Answer :
To solve the quadratic equation [tex]\(15x^2 + 13x = 0\)[/tex], we can use the quadratic formula. The general form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants. For this problem, the constants are:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Step-by-Step Solution:
1. Calculate the Discriminant:
- The discriminant ([tex]\(D\)[/tex]) is given by [tex]\(b^2 - 4ac\)[/tex].
- Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[ D = 13^2 - 4 \times 15 \times 0 = 169 - 0 = 169 \][/tex]
2. Find the Roots Using the Quadratic Formula:
- Since the discriminant is positive ([tex]\(D = 169\)[/tex]), the quadratic equation has two real roots.
3. Calculate the Roots:
- Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the square root of the discriminant ([tex]\(\sqrt{169} = 13\)[/tex]) into the quadratic formula:
[tex]\[ x_1 = \frac{-13 + \sqrt{169}}{2 \times 15} = \frac{-13 + 13}{30} = \frac{0}{30} = 0 \][/tex]
[tex]\[ x_2 = \frac{-13 - \sqrt{169}}{2 \times 15} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15} \][/tex]
Therefore, the solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
The correct answer choice is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Step-by-Step Solution:
1. Calculate the Discriminant:
- The discriminant ([tex]\(D\)[/tex]) is given by [tex]\(b^2 - 4ac\)[/tex].
- Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[ D = 13^2 - 4 \times 15 \times 0 = 169 - 0 = 169 \][/tex]
2. Find the Roots Using the Quadratic Formula:
- Since the discriminant is positive ([tex]\(D = 169\)[/tex]), the quadratic equation has two real roots.
3. Calculate the Roots:
- Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the square root of the discriminant ([tex]\(\sqrt{169} = 13\)[/tex]) into the quadratic formula:
[tex]\[ x_1 = \frac{-13 + \sqrt{169}}{2 \times 15} = \frac{-13 + 13}{30} = \frac{0}{30} = 0 \][/tex]
[tex]\[ x_2 = \frac{-13 - \sqrt{169}}{2 \times 15} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15} \][/tex]
Therefore, the solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
The correct answer choice is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
