Question

write a quadratic equation in standard form with roots -6 and 1/4

Answer 1

To write a quadratic equation in standard form given the roots, we can use the fact that if \( r_1 \) and \( r_2 \) are the roots of the quadratic equation, the equation can be expressed as:

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

In this case, the roots are \( r_1 = -6 \) and \( r_2 = \frac{1}{4} \).

Substituting the roots into the formula, we get:

\[
y = a(x + 6)\left(x - \frac{1}{4}\right)
\]

To eliminate the coefficient \( a \), we can set \( a = 1 \).

Now, we can expand the equation:

\[
y = (x + 6)\left(x - \frac{1}{4}\right)
\]

First, expand the two binomials:

\[
= x^2 - \frac{1}{4}x + 6x - \frac{6}{4}
\]
\[
= x^2 + \frac{23}{4}x - \frac{3}{2}
\]

Then, we can multiply through by 4 to eliminate the fractions:

\[
4y = 4x^2 + 23x - 6
\]

Rearranging it into standard form gives:

\[
4x^2 + 23x - 6 = 0
\]

Therefore, the quadratic equation in standard form with roots -6 and \( \frac{1}{4} \) is:

\[
4x^2 + 23x - 6 = 0
\]