Math Snap

STEP 1

1. The quadratic equation is $x^2 - 4x + k = 0$.
2. The roots of the quadratic are consecutive odd integers.
3. The sum and product of the roots can be used to find the roots and the value of $k$.

STEP 2

1. Define the roots as consecutive odd integers.
2. Use the sum of the roots to find a relationship.
3. Use the product of the roots to find the value of $k$.
4. Solve for the roots.

STEP 3

Define the roots. Let the roots be $r$ and $r+2$, where $r$ is an odd integer. This is because consecutive odd integers differ by 2.

STEP 4

Use the sum of the roots. The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $-\frac{b}{a}$. For our equation, this is:
$$r + (r + 2) = 4$$

STEP 5

Simplify the equation from the sum of the roots:
$$2r + 2 = 4$$

STEP 6

Solve for $r$:
$$2r + 2 = 4$$ $$2r = 2$$ $$r = 1$$

STEP 7

Use the product of the roots. The product of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $\frac{c}{a}$. For our equation, this is:
$$r(r + 2) = k$$

STEP 8

Substitute $r = 1$ into the product equation to find $k$:
$$1(1 + 2) = k$$ $$k = 3$$

Now that we have $r = 1$, the roots are $r$ and $r+2$:
$$r = 1$$ $$r + 2 = 3$$ The roots are $1$ and $3$.
The two consecutive odd roots of the quadratic equation are:
$$\boxed{1 \text{ and } 3}$$