a, b, c are real numbers. The quadratic equation ax2 – bx + c = 0 has equal roots, which is β, then
β3 = bc/(2a2)
To solve this problem, we need to understand the properties of a quadratic equation that has equal roots. The given quadratic equation is \(ax^2 - bx + c = 0\), where a, b, and c are real numbers, and the equal root is denoted by \(\beta\).
Quadratic Equation and Equal Roots Condition
For a general quadratic equation in the form \(Ax^2 + Bx + C = 0\), the nature of its roots is determined by its discriminant, \(D = B^2 - 4AC\).
- If \(D > 0\), the equation has two distinct real roots.
- If \(D = 0\), the equation has two equal real roots.
- If \(D < 0\), the equation has two complex conjugate roots.
In our given quadratic equation, \(ax^2 - bx + c = 0\):
- The coefficient of \(x^2\) is \(A = a\).
- The coefficient of \(x\) is \(B = -b\).
- The constant term is \(C = c\).
Since the equation has equal roots, the discriminant must be zero:
$$D = B^2 - 4AC = 0$$
Substituting the coefficients from our equation:
$$(-b)^2 - 4(a)(c) = 0$$
$$b^2 - 4ac = 0$$
Therefore, we get an important relationship: \(b^2 = 4ac\).
Root Value \(\beta\) of the Quadratic Equation
When a quadratic equation \(Ax^2 + Bx + C = 0\) has equal roots, the single root \(\beta\) can be found using the formula:
$$\beta = \frac{-B}{2A}$$
Using the coefficients from our given equation \(ax^2 - bx + c = 0\), where \(A = a\) and \(B = -b\):
$$\beta = \frac{-(-b)}{2a}$$
Thus, we find the value of the root: \(\beta = \frac{b}{2a}\).
Analyzing the Options for the Quadratic Equation
Now we will evaluate each given option using the relationships we derived: \(b^2 = 4ac\) and \(\beta = \frac{b}{2a}\).
| Option | Expression | Analysis | Result |
|---|---|---|---|
| 1 | \(\beta = b/a\) | We derived \(\beta = b/(2a)\). This option is incorrect as it is missing the factor of 2 in the denominator. | Incorrect |
| 2 | \(\beta^2 = ac\) |
We know \(\beta = b/(2a)\). Squaring both sides, we get:
$$\beta^2 = \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}$$ Now, substitute the condition for equal roots, \(b^2 = 4ac\), into the expression for \(\beta^2\):$$\beta^2 = \frac{4ac}{4a^2} = \frac{c}{a}$$ So, \(\beta^2 = c/a\). This does not match \(ac\), hence this option is incorrect. |
Incorrect |
| 3 | \(\beta^3 = bc/(2a^2)\) |
Let's calculate \(\beta^3\) using our derived value \(\beta = b/(2a)\):
$$\beta^3 = \left(\frac{b}{2a}\right)^3 = \frac{b^3}{8a^3}$$ Next, let's simplify the expression \(bc/(2a^2)\) given in the option. We can use the condition \(b^2 = 4ac\) to express \(c\). From \(b^2 = 4ac\), we can write \(c = \frac{b^2}{4a}\). Substitute this value of \(c\) into the expression \(bc/(2a^2)\):$$\frac{bc}{2a^2} = \frac{b \cdot \left(\frac{b^2}{4a}\right)}{2a^2} = \frac{\frac{b^3}{4a}}{2a^2} = \frac{b^3}{4a \cdot 2a^2} = \frac{b^3}{8a^3}$$ This derived expression for \(bc/(2a^2)\) exactly matches our calculated value for \(\beta^3\). Therefore, this option is correct. |
Correct |
| 4 | \(b^2 \ne 4ac\) | For the quadratic equation to have equal roots, the discriminant must be zero. This means \(b^2 - 4ac = 0\), which simplifies to \(b^2 = 4ac\). This option states the opposite condition, making it incorrect. | Incorrect |
Conclusion for the Quadratic Equation Problem
Based on our detailed analysis of the properties of a quadratic equation with equal roots, the relationship that holds true for \(\beta\) is \(\beta^3 = bc/(2a^2)\).
Important Questions from Numerical Reasoning
A traveller to the town reaches a crossroad. Upon asking residents A, B and C for directions to a certain destination, he gets the following responses
A: turn left
B: do not turn left
C: go straight
If only one among A, B and C is truthful, the traveller
In a city, each person has at least one hair on his/her head. At least two persons in this city are guaranteed to have exactly the same number of hair on their heads if the population of the city
S, M, E and F are working in shifts in a team to finish a project. M works with twice the efficiency of others but for half as many days as E worked. S and M have 6 hour shifts in a day, whereas E and F have 12 hours shifts. What is the ratio of contribution of M to contribution of E in the project?
If x>y>1, which of the following must be true?
(i) In x > In y
(ii) ex > ey
(iii) y2 > x2
(iv) cos x > cos y- Operators ∎, Δ and → are defined by : \(a ∎ b = \frac{{{\rm{a}} - {\rm{b}}}}{{{\rm{a}} + {\rm{b}}}};{\rm{a\;\Delta \;b}} = \frac{{{\rm{a}} + {\rm{b}}}}{{{\rm{a}} - {\rm{b}}}};{\rm{a}} \to {\rm{b}} = {\rm{ab}}.\) find the value of (66 ∎ 6) → (66 Δ 6)
