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What is the value of 'q' if one root of the equation $x^2 + px + 12 = 0$ is 4 and the equation $x^2 + px + q = 0$ has equal roots?
What is the value of 'q' if one root of the equation $x^2 + px + 12 = 0$ is 4 and the equation $x^2 + px + q = 0$ has equal roots?
For which values of m does the quadratic polynomial $P(x) = x^2 + (m + 5)x + (5m + 1)$ become a perfect square?
For which values of m does the quadratic polynomial $P(x) = x^2 + (m + 5)x + (5m + 1)$ become a perfect square?
What is the required value of $a + b$ if the equations $x - 5x + 5 = 0$ and $x + ax + bx + 5 = 0$ have a common root?
What is the required value of $a + b$ if the equations $x - 5x + 5 = 0$ and $x + ax + bx + 5 = 0$ have a common root?
If the equation $ax^2 + bx + c = 0$ has distinct real roots that are both negative, what must be true about the signs of a, b, and c?
If the equation $ax^2 + bx + c = 0$ has distinct real roots that are both negative, what must be true about the signs of a, b, and c?
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For which range of values does the quadratic $P(x) = x^2 - (2 - p)x + p - 2$ assume both positive and negative values?
For which range of values does the quadratic $P(x) = x^2 - (2 - p)x + p - 2$ assume both positive and negative values?
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What is the least value of 'a' for which the sum of the squares of the roots of the equation $x^2 - (a - 2)x - a - 1 = 0$ is minimized?
What is the least value of 'a' for which the sum of the squares of the roots of the equation $x^2 - (a - 2)x - a - 1 = 0$ is minimized?
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What is the correct interval for x such that the expression $y = 8x - x^2 - 15$ is negative?
What is the correct interval for x such that the expression $y = 8x - x^2 - 15$ is negative?
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For the quadratic polynomial $f(x) = ax^2 + bx + 8$ to be symmetric about the line $x = 2$, what is the value of $2a - b$ if the minimum value of $f(x)$ is 6?
For the quadratic polynomial $f(x) = ax^2 + bx + 8$ to be symmetric about the line $x = 2$, what is the value of $2a - b$ if the minimum value of $f(x)$ is 6?
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For which of the following sets of values of 'a' will ƒ(x) = ax + 2x(1 – a) – 4 be negative for exactly three integral values of x?
For which of the following sets of values of 'a' will ƒ(x) = ax + 2x(1 – a) – 4 be negative for exactly three integral values of x?
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Which equation describes the roots a, b, g, d given the polynomial x – bx + 3 = 0?
Which equation describes the roots a, b, g, d given the polynomial x – bx + 3 = 0?
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What is the relationship satisfied by f(x) = x2 - 6x + 5 if 1 < x < 2?
What is the relationship satisfied by f(x) = x2 - 6x + 5 if 1 < x < 2?
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What is the modulus of difference of the roots of the equation x2 + 3x – k = 0 compared to the equation x2 + 3x – 10 = 0?
What is the modulus of difference of the roots of the equation x2 + 3x – k = 0 compared to the equation x2 + 3x – 10 = 0?
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For the quadratic equations x2 – 11x + m = 0 and x2 – 14x + 2m = 0 to have a common root, what must be the value of m?
For the quadratic equations x2 – 11x + m = 0 and x2 – 14x + 2m = 0 to have a common root, what must be the value of m?
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Which of the following conditions must hold if the equations x2 + bx + c = 0 and bx2 + cx + 1 = 0 have a common root?
Which of the following conditions must hold if the equations x2 + bx + c = 0 and bx2 + cx + 1 = 0 have a common root?
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What can be said about the relationship between the discriminants of equations with roots a + d and b + d?
What can be said about the relationship between the discriminants of equations with roots a + d and b + d?
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If K1 and K2 are values for which the roots a, b satisfy (a/b) + (b/a) = 4/5, what is the correct expression for (K1/K2) + (K2/K1)?
If K1 and K2 are values for which the roots a, b satisfy (a/b) + (b/a) = 4/5, what is the correct expression for (K1/K2) + (K2/K1)?
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For the equation $x^2 + 2(a + b + c)x + 3l(ab + bc + ca) = 0$ to have real roots, which condition on l must hold?
For the equation $x^2 + 2(a + b + c)x + 3l(ab + bc + ca) = 0$ to have real roots, which condition on l must hold?
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If the equations $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root, what is the value of $a^3 + b^3 + c^3$ assuming $a ≠ 0$?
If the equations $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root, what is the value of $a^3 + b^3 + c^3$ assuming $a ≠ 0$?
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Given that $a$ and $b$ are roots of the equation $x^2 + bx + c = 0$ with $c < 0 < b$, what can be inferred about the relationship of $a$ and $b$?
Given that $a$ and $b$ are roots of the equation $x^2 + bx + c = 0$ with $c < 0 < b$, what can be inferred about the relationship of $a$ and $b$?
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If $b > a$, which statement about the roots of the equation $(x - a)(x - b) - 1 = 0$ is accurate?
If $b > a$, which statement about the roots of the equation $(x - a)(x - b) - 1 = 0$ is accurate?
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Which expression does NOT hold good based on the graph of $y = ax^2 + bx + c$?
Which expression does NOT hold good based on the graph of $y = ax^2 + bx + c$?
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What is the sum of the solutions for the equation $\frac{x - 2}{x - 4} + 2 = 0$ with $x > 0$?
What is the sum of the solutions for the equation $\frac{x - 2}{x - 4} + 2 = 0$ with $x > 0$?
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What is the set of all real numbers $x$ such that $x^2 - |x + 2| + x > 0$ holds true?
What is the set of all real numbers $x$ such that $x^2 - |x + 2| + x > 0$ holds true?
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If $a$ and $b$ are roots of the equation $3x^2 - 9x - l = 0$ with conditions $1 < a < 3$ and $3 < b < 5$, what range must $l$ lie within?
If $a$ and $b$ are roots of the equation $3x^2 - 9x - l = 0$ with conditions $1 < a < 3$ and $3 < b < 5$, what range must $l$ lie within?
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What is the value of $\frac{a_{10} - 2a_{8}}{2a_{9}}$ given that $a_n = a_n - b_n$ for $n \geq 1$?
What is the value of $\frac{a_{10} - 2a_{8}}{2a_{9}}$ given that $a_n = a_n - b_n$ for $n \geq 1$?
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For which value of $b$ do the equations $x^2 + bx - 1 = 0$ and $x^2 + x + b = 0$ have one root in common?
For which value of $b$ do the equations $x^2 + bx - 1 = 0$ and $x^2 + x + b = 0$ have one root in common?
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Which interval is a subset of the set $S$ for values of $a$ such that the quadratic equation $ax^2 - x + a = 0$ has roots $x_1$ and $x_2$ satisfying $|x_1 - x_2| < 1$?
Which interval is a subset of the set $S$ for values of $a$ such that the quadratic equation $ax^2 - x + a = 0$ has roots $x_1$ and $x_2$ satisfying $|x_1 - x_2| < 1$?
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If $a_4 = 28$, what is the value of $p + 2q$?
If $a_4 = 28$, what is the value of $p + 2q$?
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What is the value of $a_{12}$ if $a_n = pa_n + qb_n$?
What is the value of $a_{12}$ if $a_n = pa_n + qb_n$?
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Which of the following conditions must be satisfied for the quadratic equation $ax^2 - x + a = 0$ to have two distinct real roots?
Which of the following conditions must be satisfied for the quadratic equation $ax^2 - x + a = 0$ to have two distinct real roots?
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What is the relationship between the roots $a$ and $b$ in the equation $x^2 - x - 1 = 0$ given that $a \neq b$?
What is the relationship between the roots $a$ and $b$ in the equation $x^2 - x - 1 = 0$ given that $a \neq b$?
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What is one necessary condition for the roots $x_1$ and $x_2$ of the quadratic equation $ax^2 - x + a = 0$ to satisfy the inequality $|x_1 - x_2| < 1$?
What is one necessary condition for the roots $x_1$ and $x_2$ of the quadratic equation $ax^2 - x + a = 0$ to satisfy the inequality $|x_1 - x_2| < 1$?
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What is the value of $c$ in the equation $3b = \frac{1}{2}$?
What is the value of $c$ in the equation $3b = \frac{1}{2}$?
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In the equation $acx^2 - bx + 1 = 0$, what is the term that adjusts the linear coefficient?
In the equation $acx^2 - bx + 1 = 0$, what is the term that adjusts the linear coefficient?
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Which of the following represents a valid range for $x$ extricated from $x \in (3, \infty)$?
Which of the following represents a valid range for $x$ extricated from $x \in (3, \infty)$?
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From the expression $x^2 - 2(p^2 - 2q)x + p^2(p^2 - 4q) = 0$, which term is key to determining the quadratic's properties?
From the expression $x^2 - 2(p^2 - 2q)x + p^2(p^2 - 4q) = 0$, which term is key to determining the quadratic's properties?
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Which expression signifies a quadratic function derived from the variable transformation when $p$ influences $q$?
Which expression signifies a quadratic function derived from the variable transformation when $p$ influences $q$?
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What is the solution for $x$ when it lies in the interval $[-2,3)$?
What is the solution for $x$ when it lies in the interval $[-2,3)$?
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In the equation $x \in R - (0, 1]$, which of the following intervals is NOT included in the solution?
In the equation $x \in R - (0, 1]$, which of the following intervals is NOT included in the solution?
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Which of the following cases correctly represents range $[20/9, 4) \cup (5, \infty)$?
Which of the following cases correctly represents range $[20/9, 4) \cup (5, \infty)$?
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Study Notes
Quadratic Equations and Their Roots
- Common Root: If two quadratic equations have a common root, then the value of the common root can be found by solving the system of equations.
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Discriminant: The discriminant of a quadratic equation ax² + bx + c = 0 is b² - 4ac. It helps determine the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one real root (a double root).
- If b² - 4ac < 0, the equation has two complex roots.
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Sum and Product of Roots: For a quadratic equation ax² + bx + c = 0,
- The sum of the roots is -b/a.
- The product of the roots is c/a.
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Quadratic Formula: The roots of the quadratic equation ax² + bx + c = 0 are given by the quadratic formula:
- x = (-b ± √(b² - 4ac)) / 2a
- Perfect Square: A quadratic expression is a perfect square if it can be factored as (ax + b)² or (ax - b)².
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Inequalities:
- To find the solution set for an inequality involving a quadratic expression, analyze the sign of the expression in different intervals defined by the roots of the quadratic equation.
- If the leading coefficient of the quadratic expression is positive, the expression is positive outside the interval defined by the roots and negative inside the interval.
- If the leading coefficient of the quadratic expression is negative, the expression is negative outside the interval defined by the roots and positive inside the interval.
Specific Equation Solving Techniques
- Sum and Product of Roots for Solving Systems: Use the sum and product of roots relationships to solve systems of equations involving quadratic equations.
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Conditions for Specific Root Relationships:
- Equal Roots: For a quadratic equation ax² + bx + c = 0, the roots are equal if and only if b² - 4ac = 0.
- Distinct Real Roots: The quadratic ax² + bx + c = 0 has distinct real roots if and only if b² - 4ac > 0.
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Negative Roots: The quadratic equation ax² + bx + c = 0 has both roots negative if:
- a, b, and c are all of the same sign (positive or negative)
- The discriminant b² - 4ac is positive
- -b/a (the sum of the roots) is negative
- c/a (the product of the roots ) is positive
- Factoring for Solution: If possible, factor the quadratic expression to find the roots.
Additional Notes
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Graphs: The graph of a quadratic function is a parabola. The shape of the parabola depends on the sign of the leading coefficient.
- If the leading coefficient is positive, the parabola opens upwards.
- If the leading coefficient is negative, the parabola opens downwards.
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Vertex: The vertex of a parabola is the point where the graph changes direction.
- The x-coordinate of the vertex is -b/2a.
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Minimum/Maximum Values: The minimum or maximum value of a quadratic function is attained at the vertex.
- The function has a minimum value if the parabola opens upwards.
- The function has a maximum value if the parabola opens downwards.
- Symmetry: The graph of a quadratic function is symmetric about the vertical line passing through the vertex.
- Transformations: Quadratic functions can be transformed by shifting, stretching, and reflecting their graphs.
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Description
Explore the fundamental concepts of quadratic equations including common roots, discriminants, and the quadratic formula. This quiz covers the nature of roots and their sums and products, ensuring a comprehensive understanding of the topic. Test your knowledge and problem-solving skills regarding various aspects of quadratic equations.
