d We can write a quadratic equation in factored form using the given roots. Then we will change it to standard form by multiplying the factors.

Factored Form : Standard Form : a (x - p) (x - q) = 0 a x^{2} + b x + c = 0

In the factored form,

p

and

q

are the roots of the equation. Since we are told the roots are

\frac{2}{3}

and

- \frac{5}{7},

we can partially write the factored form of our equation.

a (x - \frac{2}{3}) (x - (- \frac{5}{7})) = 0 ⇕ a (x - \frac{2}{3}) (x + \frac{5}{7}) = 0

Since

a

does not have any effect on the roots, we can choose any value. For simplicity and in order to have integer coefficients, we will let

a = 21 .

This is a common multiple of both denominators of the given roots and will allow us to eliminate the fractions when we distribute.

21 (x - \frac{2}{3}) (x + \frac{5}{7}) = 0

Finally, let's use the Distributive Property to obtain the standard form.

21 (x - \frac{2}{3}) (x + \frac{5}{7}) = 0

▼

Distribute

21

Split into factors

3 (7) (x - \frac{2}{3}) (x + \frac{5}{7}) = 0

Commutative Property of Multiplication

3 (x - \frac{2}{3}) (7) (x + \frac{5}{7}) = 0

Distribute $3$

(3 x - 2) (7) (x + \frac{5}{7}) = 0

Distribute $7$

(3 x - 2) (7 x + 5) = 0

▼

Multiply parentheses

Distribute $(7 x + 5)$

3 x (7 x + 5) - 2 (7 x + 5) = 0

Distribute $3 x$

21 x^{2} + 15 x - 2 (7 x + 5) = 0

Distribute $- 2$

21 x^{2} + 15 x - 14 x - 10 = 0

Subtract terms

21 x^{2} + x - 10 = 0

Please note that this is just one example of a quadratic function that satisfies the given requirements.

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