Math / Algebra

QuestionQUESTION 5 / 10 - LEVEL 2 10 MARK(S)
The quadratic equating $x^{2}-4 x+k=0$ has two consecutive odd roots. What are the two roots?

Studdy Solution

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 $k$ :
$1(1 + 2) = k$ $k = 3$

STEP 9

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:
$\boxed{1 \text{ and } 3}$

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QuestionQUESTION 5 / 10 - LEVEL 2 10 MARK(S)
The quadratic equating $x^{2}-4 x+k=0$ has two consecutive odd roots. What are the two roots?

Studdy Solution

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 $k$ :
$1(1 + 2) = k$ $k = 3$

STEP 9

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:
$\boxed{1 \text{ and } 3}$

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QuestionQUESTION 5 / 10 - LEVEL 2 10 MARK(S)The quadratic equating x2−4x+k=0x^{2}-4 x+k=0x2−4x+k=0 has two consecutive odd roots. What are the two roots?

Studdy Solution

STEP 1

1. The quadratic equation is x2−4x+k=0 x^2 - 4x + k = 0 x2−4x+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 k 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 k k.4. Solve for the roots.

STEP 3

Define the roots. Let the roots be r r r and r+2 r+2 r+2, where r r 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 ax2+bx+c=0 ax^2 + bx + c = 0 ax2+bx+c=0 is given by −ba -\frac{b}{a} −ab​. For our equation, this is:r+(r+2)=4 r + (r + 2) = 4 r+(r+2)=4

STEP 5

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

STEP 6

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

STEP 7

Use the product of the roots. The product of the roots of a quadratic equation ax2+bx+c=0 ax^2 + bx + c = 0 ax2+bx+c=0 is given by ca \frac{c}{a} ac​. For our equation, this is:r(r+2)=k r(r + 2) = k r(r+2)=k

STEP 8

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

STEP 9

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

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QuestionQUESTION 5 / 10 - LEVEL 2 10 MARK(S)
The quadratic equating $x^{2}-4 x+k=0$ has two consecutive odd roots. What are the two roots?

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 $k$ .

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.

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.

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:
$r + (r + 2) = 4$

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

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

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:
$r(r + 2) = k$

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:
$\boxed{1 \text{ and } 3}$