Find the value of 𝑝 for which one
root of the quadratic equation 𝑝𝑥 squared minus 14𝑥 plus eight equals zero is six
times the other.
We’re looking for the roots. Let’s call one of the roots 𝑥. One of the roots is six times the
other. Our second root we can then call
six times 𝑥. We also remember some properties of
For 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐,
the sum of the roots equals negative 𝑏 over 𝑎. And the product of the roots equals
𝑐 over 𝑎. Our equation is 𝑝𝑥 squared minus
14𝑥 plus eight equals zero. The value of 𝑎 in our equation is
𝑝. The value of 𝑏 equals negative
14. The value of 𝑐 is positive
eight. And that means the sum of the roots
of our equation all equal the negative of negative 14 over 𝑝. The product of our roots equals
eight over 𝑝.
Let’s take the sum of the roots and
set them equal to 𝑥 plus six 𝑥. Remember that we’ve already said
our roots are 𝑥 and six 𝑥. We can also take the product of the
roots, eight over 𝑝, and set that equal to 𝑥 times six 𝑥. Working with the sum, 𝑥 plus six
𝑥 equals seven 𝑥.
Now we have seven 𝑥 equals
14𝑝. We’ll get 𝑥 by itself here, and we
do that by multiplying by one over seven on both sides of the equation. On the left, the sevens cancel out,
leaving us with 𝑥. On the right, we have 14 over seven
𝑝. We know that 14 over seven can be
simplified to two. 𝑥 is equal to two 𝑝.
Now we can take this two 𝑝 value
and plug it in for 𝑥. And this is what we would have: two
over 𝑝 times six times two over 𝑝. Since we’re dealing with
multiplication, we can multiply the numerators together. Two times six times two equals
24. We also multiply the denominators
together. 𝑝 times 𝑝 equals 𝑝 squared.
Our new equation looks like this:
24 over 𝑝 squared equals eight over 𝑝. To solve for 𝑝, I want 𝑝 out of
the denominator. I do that by multiplying both sides
of the equation by 𝑝 squared over one. On the left, everything cancels
out, and we’re left with 24. And on the right, we have this:
eight times 𝑝 squared over 𝑝. Since we’re dealing with
multiplication, we can break it into two parts: eight over one times 𝑝 squared over
𝑝. 𝑝 squared over 𝑝 to the first
power can be simplified to 𝑝.
Our new statement says 24 equals
eight 𝑝. From there, we divide both sides of
the equation by eight. On the right, eight divided by
eight cancels out. 24 divided by eight equals
three. 𝑝 equals three. The equation three 𝑥 squared minus
14𝑥 plus eight has a root that is six times the other root.