Math Snap

STEP 1

1. The roots of the quadratic equation are given as mixed numbers.
2. The standard form of a quadratic equation is $ax^2 + bx + c = 0$.
3. To form a quadratic equation from its roots, use the relationship between roots and coefficients.

STEP 2

1. Convert the given roots into improper fractions.
2. Use the sum and product of roots to form the quadratic equation.
3. Form a new equation with roots that are twice the original roots.

STEP 3

Convert the mixed numbers to improper fractions. The roots given are $3 \frac{1}{2}$ and $-2 \frac{1}{3}$.
$$3 \frac{1}{2} = \frac{7}{2}$$ $$-2 \frac{1}{3} = -\frac{7}{3}$$

STEP 4

Use the sum and product of the roots to form the quadratic equation. If the roots are $r_1$ and $r_2$, then the quadratic equation is:
$$x^2 - (r_1 + r_2)x + r_1r_2 = 0$$ Calculate the sum and product of the roots:
Sum:
$$r_1 + r_2 = \frac{7}{2} + \left(-\frac{7}{3}\right) = \frac{21}{6} - \frac{14}{6} = \frac{7}{6}$$ Product:
$$r_1 \cdot r_2 = \frac{7}{2} \cdot \left(-\frac{7}{3}\right) = -\frac{49}{6}$$

STEP 5

Substitute the sum and product into the quadratic equation formula:
$$x^2 - \left(\frac{7}{6}\right)x - \frac{49}{6} = 0$$ Multiply through by 6 to clear the fractions:
$$6x^2 - 7x - 49 = 0$$

STEP 6

Form an equation whose roots are twice those in the previous part. If the original roots are $r_1$ and $r_2$, the new roots are $2r_1$ and $2r_2$.
Calculate the new sum and product:
New Sum:
$$2r_1 + 2r_2 = 2(r_1 + r_2) = 2 \times \frac{7}{6} = \frac{7}{3}$$ New Product:
$$(2r_1)(2r_2) = 4(r_1r_2) = 4 \times \left(-\frac{49}{6}\right) = -\frac{196}{6} = -\frac{98}{3}$$

Substitute the new sum and product into the quadratic equation formula:
$$x^2 - \left(\frac{7}{3}\right)x - \frac{98}{3} = 0$$ Multiply through by 3 to clear the fractions:
$$3x^2 - 7x - 98 = 0$$ The quadratic equations are:
(a) $6x^2 - 7x - 49 = 0$
(b) $3x^2 - 7x - 98 = 0$