If p and q (p > q) are the roots of the equation x 2 - 60x + 899 = 0, then which one of the following is correct ?
p - 2q + 27 = 0
Solving Quadratic Equation to Find Roots p and q
The problem asks us to find which relationship is correct for the roots, p and q (with p > q), of the quadratic equation \(x^2 - 60x + 899 = 0\).
To solve this, we first need to find the roots of the given quadratic equation. A standard quadratic equation is in the form \(ax^2 + bx + c = 0\). Comparing this with our equation, \(x^2 - 60x + 899 = 0\), we have:
- a = 1
- b = -60
- c = 899
We can find the roots using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
First, let's calculate the discriminant, \(\Delta = b^2 - 4ac\):
\[\Delta = (-60)^2 - 4(1)(899)\]
\[\Delta = 3600 - 3596\]
\[\Delta = 4\]
Now, substitute the values of a, b, and the discriminant into the quadratic formula to find the roots:
\[x = \frac{-(-60) \pm \sqrt{4}}{2(1)}\]
\[x = \frac{60 \pm 2}{2}\]
This gives us two possible roots:
- \(x_1 = \frac{60 + 2}{2} = \frac{62}{2} = 31\)
- \(x_2 = \frac{60 - 2}{2} = \frac{58}{2} = 29\)
The problem states that p and q are the roots and \(p > q\). Therefore, we assign the larger root to p and the smaller root to q:
- p = 31
- q = 29
Now we need to check which of the given options is correct by substituting the values of p and q.
Checking the Relationships Between Roots p and q
Let's test each option:
- Option 1: \(p - q - 1 = 0\)
- Substitute p=31 and q=29: \(31 - 29 - 1 = 2 - 1 = 1\).
- Is \(1 = 0\)? No. Option 1 is incorrect.
- Option 2: \(p - 2q + 27 = 0\)
- Substitute p=31 and q=29: \(31 - 2(29) + 27 = 31 - 58 + 27\).
- Calculate: \(31 - 58 + 27 = -27 + 27 = 0\).
- Is \(0 = 0\)? Yes. Option 2 is correct.
- Option 3: \(2p - q - 30 = 0\)
- Substitute p=31 and q=29: \(2(31) - 29 - 30 = 62 - 29 - 30\).
- Calculate: \(62 - 29 - 30 = 33 - 30 = 3\).
- Is \(3 = 0\)? No. Option 3 is incorrect.
- Option 4: \(3p - 2q - 43 = 0\)
- Substitute p=31 and q=29: \(3(31) - 2(29) - 43 = 93 - 58 - 43\).
- Calculate: \(93 - 58 - 43 = 35 - 43 = -8\).
- Is \(-8 = 0\)? No. Option 4 is incorrect.
Therefore, the correct relationship between the roots p and q is given by Option 2: \(p - 2q + 27 = 0\).
Revision Table: Solving Quadratic Equations
| Concept | Description | Formula/Method |
|---|---|---|
| Quadratic Equation | An equation of the form \(ax^2 + bx + c = 0\), where \(a \ne 0\). | \(ax^2 + bx + c = 0\) |
| Roots of Equation | The values of x that satisfy the equation. | Solve for x |
| Discriminant (\(\Delta\)) | Determines the nature of the roots. | \(\Delta = b^2 - 4ac\) |
| Quadratic Formula | Used to find the roots of a quadratic equation. | \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\) |
Additional Information on Quadratic Equation Roots
Besides the quadratic formula, the properties of the roots of a quadratic equation are also described by Vieta's formulas. For a quadratic equation \(ax^2 + bx + c = 0\) with roots p and q, Vieta's formulas state:
- Sum of roots: \(p + q = -\frac{b}{a}\)
- Product of roots: \(pq = \frac{c}{a}\)
In our case, for \(x^2 - 60x + 899 = 0\):
- Sum of roots: \(p + q = -(\frac{-60}{1}) = 60\)
- Product of roots: \(pq = \frac{899}{1} = 899\)
Using \(p=31\) and \(q=29\):
- \(p + q = 31 + 29 = 60\). This matches the sum of roots formula.
- \(pq = 31 \times 29\). Let's calculate \(31 \times 29 = 31 \times (30 - 1) = 31 \times 30 - 31 \times 1 = 930 - 31 = 899\). This matches the product of roots formula.
Vieta's formulas provide an alternative way to check the roots once they are found or to analyze the relationship between roots without explicitly solving the equation.
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