To find the quadratic equation given its roots, we can follow these steps:
1. Identify the Roots:
The roots of the quadratic equation are [tex]\( x = -\frac{2}{3} \)[/tex] and [tex]\( x = 4 \)[/tex].
2. Form the Factorized Equation:
Using the roots, we can write the quadratic equation in its factorized form:
[tex]\[
\left( x - \left(-\frac{2}{3}\right) \right) \left( x - 4 \right) = 0
\][/tex]
This simplifies to:
[tex]\[
\left( x + \frac{2}{3} \right) \left( x - 4 \right) = 0
\][/tex]
3. Clear the Fractions:
To eliminate the fraction, multiply each term by 3:
[tex]\[
3 \left( x + \frac{2}{3} \right) \left( x - 4 \right) = 0
\][/tex]
Hence, we get:
[tex]\[
(3x + 2) \left( x - 4 \right) = 0
\][/tex]
4. Expand the Product:
Next, we expand the equation:
[tex]\[
(3x + 2)(x - 4) = 3x^2 - 12x + 2x - 8
\][/tex]
Simplify by combining like terms:
[tex]\[
3x^2 - 10x - 8 = 0
\][/tex]
5. Write the Quadratic Equation:
The quadratic equation is:
[tex]\[
3x^2 - 10x - 8 = 0
\][/tex]
Therefore, the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[
a = 3, \quad b = -10, \quad c = -8
\][/tex]
So the quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[
3x^2 - 10x - 8 = 0
\][/tex]
