Question

Turn around with the value is -1 and 7/3 are solutions to the quadratic equation 3x - 4x - 4 = 3

Answer 1

It looks like you want to find the quadratic equation based on the given solutions, which are \( -1 \) and \( \frac{7}{3} \).

Here's how to form the quadratic equation from its roots:

If \( r_1 \) and \( r_2 \) are the roots (solutions) of the quadratic equation, the quadratic can be expressed in the form:

\[
f(x) = a(x - r_1)(x - r_2)
\]

Where \( a \) is a non-zero constant. Since the roots provided are \( r_1 = -1 \) and \( r_2 = \frac{7}{3} \), we can plug these into the formula:

\[
f(x) = a(x - (-1))(x - \frac{7}{3}) = a(x + 1)(x - \frac{7}{3})
\]

To eliminate the fraction, we can multiply by 3 (the denominator):

\[
f(x) = a \cdot 3(x + 1)(3x - 7)
\]

Expanding the expression:

\[
f(x) = a \cdot 3[(x + 1)(3x - 7)] \\
= a \cdot 3[3x^2 - 7x + 3x - 7] \\
= a \cdot 3[3x^2 - 4x - 7]
\]

For simplicity, let \( a = 1 \):

\[
f(x) = 3x^2 - 4x - 7
\]

Now, we can use your original equation:

\[
3x - 4x - 4 = 3
\]
This simplifies to:

\[
-4x - 1 = 0 \quad (after moving everything to the left)
\]
Does not match the derived quadratic.

Instead, the correct quadratic form is
\[
3x^2 - 4x - 7 = 0
\]
This should match your requirement.

Thus, the quadratic equation with solutions \( -1 \) and \( \frac{7}{3} \) is:

\[
3x^2 - 4x - 7 = 0
\]