Video: CBSE Class X • Pack 2 • 2017 • Question 5

CBSE Class X • Pack 2 • 2017 • Question 5

03:55

Video Transcript

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 quadratic roots.

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.

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