If 2p + q = 19 and 8p3 + q3 = 361, then find the value of pq.
57
Finding the Value of pq from Given Algebraic Equations
We are given two equations involving the variables p and q:
- Equation 1: \(2p + q = 19\)
- Equation 2: \(8p^3 + q^3 = 361\)
We need to find the value of the product \(pq\).
The second equation involves cubic terms, which suggests using an algebraic identity related to cubes. The identity for the cube of a sum, \((a+b)^3\), is useful here:
\((a+b)^3 = a^3 + b^3 + 3ab(a+b)\)
Let's apply this identity to the term \((2p + q)^3\). We can consider \(a = 2p\) and \(b = q\).
So, substituting these into the identity:
\((2p + q)^3 = (2p)^3 + q^3 + 3(2p)(q)(2p+q)\)
Simplifying the terms:
\((2p + q)^3 = 8p^3 + q^3 + 6pq(2p+q)\)
Now, we can substitute the values given in the problem into this expanded equation. From Equation 1, we know that \((2p + q) = 19\). From Equation 2, we know that \((8p^3 + q^3) = 361\).
Substituting these values:
\(19^3 = 361 + 6pq(19)\)
Next, let's calculate the value of \(19^3\).
\(19^2 = 19 \times 19 = 361\)
\(19^3 = 19^2 \times 19 = 361 \times 19\)
Performing the multiplication:
| Calculation | Result |
|---|---|
| \(361 \times 19\) | \(6859\) |
So, the equation becomes:
\(6859 = 361 + 114pq\)
Our goal is to find the value of \(pq\). We need to rearrange the equation to isolate the \(pq\) term.
Subtract 361 from both sides of the equation:
\(6859 - 361 = 114pq\)
\(6498 = 114pq\)
Now, divide both sides by 114 to solve for \(pq\):
\(pq = \frac{6498}{114}\)
Performing the division:
| Division | Result |
|---|---|
| \(6498 \div 114\) | \(57\) |
Thus, the value of \(pq\) is 57.
Revision Table: Solving for pq
| Step | Action | Result/Formula |
|---|---|---|
| 1 | Identify given equations | \(2p+q=19\), \(8p^3+q^3=361\) |
| 2 | Apply \((a+b)^3\) identity with \(a=2p, b=q\) | \((2p+q)^3 = (2p)^3 + q^3 + 3(2p)q(2p+q)\) |
| 3 | Simplify the identity application | \((2p+q)^3 = 8p^3 + q^3 + 6pq(2p+q)\) |
| 4 | Substitute given values into the equation | \(19^3 = 361 + 6pq(19)\) |
| 5 | Calculate \(19^3\) | \(19^3 = 6859\) |
| 6 | Substitute \(19^3\) value | \(6859 = 361 + 114pq\) |
| 7 | Isolate \(114pq\) term | \(6859 - 361 = 114pq \implies 6498 = 114pq\) |
| 8 | Solve for \(pq\) | \(pq = \frac{6498}{114} = 57\) |
Additional Information: Useful Algebraic Identities
Algebraic identities are fundamental tools in solving equations and simplifying expressions. They provide shortcuts and standard formulas for common algebraic manipulations.
Some frequently used identities related to powers include:
- Square of a Sum: \((x+y)^2 = x^2 + 2xy + y^2\)
- Square of a Difference: \((x-y)^2 = x^2 - 2xy + y^2\)
- Difference of Squares: \(x^2 - y^2 = (x-y)(x+y)\)
- Cube of a Sum: \((x+y)^3 = x^3 + y^3 + 3xy(x+y)\)
- Cube of a Difference: \((x-y)^3 = x^3 - y^3 - 3xy(x-y)\)
- Sum of Cubes: \(x^3 + y^3 = (x+y)(x^2 - xy + y^2)\)
- Difference of Cubes: \(x^3 - y^3 = (x-y)(x^2 + xy + y^2)\)
Understanding and recognizing these identities can significantly simplify problems like the one solved above, where a relationship between linear terms and cubic terms is given.
Similar Questions
Simplify (x - y + z) 2- (x - y - z) 2.
If 27x 3 – 64y 3 = (Ax + By) (Cx 2 – Dy 2 + 12xy), then the value of 4A + B + 3C + 2D is:
If 2x 2- 8x - 1 = 0, then what is the value of \(\rm 8x^3 - \frac{1}{x^3}\) ?
If \((x^2+\frac{1}{x^2})=7\), and 0 < x < 1, find the value of \(x^2-\frac{1}{x^2} \).
If x + \(\frac{1}{x}\) = 7, then the value of x6 + \(\frac{1}{x^6}\) is:
If x2 - \(\frac{1}{x^2}\) = 4 \(\sqrt2\), what is the value of x4 - \(\frac{1}{x^4}\)?
If a - b = 8 and ab = 9, then the value of a + b is ________.
If \(a + \frac{1}{a} = 5 \) , then what is the value of \({a^3} + \frac{1}{{{a^3}}}\)?
If \(\frac{x}{8} + \frac{8}{x} = 1\), then the value of x3 is :
If abc = 5, what is the value of (\({1 \over 1 \ + \ a \ + \ b^{-1} }\) + \({1 \over 1 \ + \ b \ + \ 5 c^{-1}}\) +\(\frac{1}{1+\frac{c}{5}+a^{-1}}\) \(\) )?
Important Questions from Identities
If \(2x + { {1} \over 3x}= 5, x ≠ 0\) , then what is the value of \(27x^3+{{1} \over 8x^3}\) ?
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
If a(a + b + c) 2= 1792; b(a + b + c) 2= 1536; c(a + b + c) 2= 768 then what will be the value of a?
