Question

piper is older than ariana. their ages are consecutive odd integers. find pipers age if the sum of the square of pipers age and 3 times arianas age is 264.

Explanation:

Step1: Let the variables

Let Ariana's age be $x$. Since their ages are consecutive odd integers and Piper is older, Piper's age is $x + 2$.

Step2: Set up the equation

The sum of the square of Piper's age and 3 times Ariana's age is 264. So, $(x + 2)^2+3x=264$.

Step3: Expand the square

Expand $(x + 2)^2$ using the formula $(a + b)^2=a^{2}+2ab + b^{2}$. Here $a=x$ and $b = 2$, so $(x + 2)^2=x^{2}+4x+4$. The equation becomes $x^{2}+4x + 4+3x=264$.

Step4: Combine like - terms

Combine the $x$ terms: $x^{2}+7x+4 = 264$.

Step5: Rearrange to standard quadratic form

Subtract 264 from both sides to get $x^{2}+7x-260=0$.

Step6: Factor the quadratic equation

Factor $x^{2}+7x - 260$. We need two numbers that multiply to - 260 and add up to 7. The numbers are 20 and - 13. So, $x^{2}+7x-260=(x + 20)(x - 13)=0$.

Step7: Solve for x

Set each factor equal to zero: $x+20 = 0$ gives $x=-20$; $x - 13=0$ gives $x = 13$. Since age cannot be negative, $x = 13$.

Step8: Find Piper's age

Piper's age is $x + 2$. Substitute $x = 13$ into $x + 2$, we get $13+2=15$.

Answer:

15