a, b, c are real numbers. The quadratic equation ax² – bx + c = 0 has equal roots, which is β, then

Question

a, b, c are real numbers. The quadratic equation ax2 &ndash; bx + c = 0 has equal roots, which is &beta;, then

Prepp Team · Accepted Answer

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 $\beta^3 = bc/(2a^2)$.

Prepp Team · Answer

&beta; = b/a

Prepp Team · Answer

&beta;2 = ac

Prepp Team · Answer

b2 &ne; 4ac

Quadratic Equation and Equal Roots Condition

Root Value \(\beta\) of the Quadratic Equation

Analyzing the Options for the Quadratic Equation

Conclusion for the Quadratic Equation Problem

Important Questions from Numerical Reasoning

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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