Quadratic Equations

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What is the condition for a tangent to be parallel to the x-axis?

f'(x) = 0 (correct)

f'(x) < 0

f'(x) ≠ 0

f'(x) > 0

If α is a root of the equation f(x) = 0, what can be said about the polynomial f(x)?

f(x) is exactly divisible by (x - α)²

f(x) is exactly divisible by (x + α)

f(x) is exactly divisible by (x - α) (correct)

f(x) is exactly divisible by (x³ - α)

If the roots of the quadratic equations ax² + bx + c = 0 and a²x² + b²x + c² = 0 are the same, what can be said about b₁²/b₂²?

b₁²/b₂² = a₂c₂/a₁c₁

b₁²/b₂² = c₁/c₂

b₁²/b₂² = a₁/a₂

b₁²/b₂² = a₁c₁/a₂c₂ (correct)

If one root of the quadratic equation ax² + bx + c = 0 is k times the other root, what can be said about the equation?

(k + 1)²b² = kac Signup and view all the answers

What is the quadratic expression in x?

ax² + bx + c Signup and view all the answers

If α and β are the roots of the quadratic equation ax² + bx + c = 0, then what is the sum of the roots?

-b/a Signup and view all the answers

If α and β are the roots of the quadratic equation ax² + bx + c = 0, then what is the product of the roots?

c/a Signup and view all the answers

What is the quadratic equation whose roots are α and β?

x² - (α + β)x + αβ = 0 Signup and view all the answers

What is the value of α² + β² in terms of the coefficients of the quadratic equation ax² + bx + c = 0?

b² - 4ac Signup and view all the answers

What is the value of α³ + β³ in terms of the coefficients of the quadratic equation ax² + bx + c = 0?

b(b² - 3ac) Signup and view all the answers

If α and β are the roots of the equation x^2 - ax + b = 0, what is the value of α + β?

a Signup and view all the answers

If the roots of the equation x^2 - (α + β)x + αβ = 0 are a and b, what is the value of α and β?

α and β Signup and view all the answers

If Vn = α^n + β^n, what is the value of Vn+1?

aVn + bVn-1 Signup and view all the answers

If the roots of the equation (x - α)(x - β) + c = 0 are a and b, what is the relationship between α, β, a, and b?

α + β = a + b Signup and view all the answers

If a and b are the roots of the quadratic equation x^2 + bx + c = 0, what is the value of a^2 + b^2 in terms of the coefficients?

b^2 - 4c Signup and view all the answers

If a quadratic equation has complex roots, what can be said about the discriminant?

It is a negative number. Signup and view all the answers

What is the quadratic formula used to find?

The roots of a quadratic equation. Signup and view all the answers

If $α$ and $β$ are the roots of the equation $x^2 - ax + b = 0$, then what is the value of $α^2 + β^2$ in terms of $a$ and $b$?

$a^2 - 2b$ Signup and view all the answers

If the roots of the equation $(x - α)(x - β) + c = 0$ are $a$ and $b$, then what is the value of $c$ in terms of $a$, $b$, $α$, and $β$?

$ab - αβ$ Signup and view all the answers

If the roots of the equation $x^2 + ax + b = 0$ are $α$ and $β$, then what is the value of $αβ$ in terms of $a$ and $b$?

$b/a$ Signup and view all the answers

If α and β are the roots of the quadratic equation x² - Sx + P = 0, which of the following is true?

α + β = S and αβ = P Signup and view all the answers

If we replace x by 1/x in the quadratic equation ax² + bx + c = 0, what is the resulting equation?

cx² + bx + a = 0 Signup and view all the answers

If the quadratic equation x² - Sx + P = 0 has roots α and β, what is the equation whose roots are 1/α and 1/β?

cx² + bx + a = 0 Signup and view all the answers

If α and β are the roots of the quadratic equation x² - Sx + P = 0, what is the value of (α + β)²?

S² - 2P Signup and view all the answers

If the quadratic equation ax² + bx + c = 0 has roots α and β, what is the equation whose roots are kα and kβ?

ax² + kbx + k²c = 0 Signup and view all the answers

If α and β are the roots of the equation x² - Sx + P = 0, then what is the equation whose roots are -α and -β?

ax² - Sx - P = 0 Signup and view all the answers

If α and β are the roots of the equation ax² + bx + c = 0, then what is the equation whose roots are 1/α and 1/β?

cx² + bx + a = 0 Signup and view all the answers

If α and β are the roots of the equation x² - Sx + P = 0, then what is the value of (α - β)² in terms of S and P?

S² - 4P Signup and view all the answers

If α and β are the roots of the equation ax² + bx + c = 0, then what is the equation whose roots are α² and β²?

a(x² - bx + c) = 0 Signup and view all the answers

If α and β are the roots of the equation x² - Sx + P = 0, then what is the value of α⁴ + β⁴ in terms of S and P?

(S² + P²)² - 2P(S³ + P) Signup and view all the answers

Quadratic Equations

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Questions and Answers

What is the condition for a tangent to be parallel to the x-axis?

If α is a root of the equation f(x) = 0, what can be said about the polynomial f(x)?

If the roots of the quadratic equations ax² + bx + c = 0 and a²x² + b²x + c² = 0 are the same, what can be said about b₁²/b₂²?

If one root of the quadratic equation ax² + bx + c = 0 is k times the other root, what can be said about the equation?

What is the quadratic expression in x?

If α and β are the roots of the quadratic equation ax² + bx + c = 0, then what is the sum of the roots?

If α and β are the roots of the quadratic equation ax² + bx + c = 0, then what is the product of the roots?

What is the quadratic equation whose roots are α and β?

What is the value of α² + β² in terms of the coefficients of the quadratic equation ax² + bx + c = 0?

What is the value of α³ + β³ in terms of the coefficients of the quadratic equation ax² + bx + c = 0?

If α and β are the roots of the equation x^2 - ax + b = 0, what is the value of α + β?

If the roots of the equation x^2 - (α + β)x + αβ = 0 are a and b, what is the value of α and β?

If Vn = α^n + β^n, what is the value of Vn+1?

If the roots of the equation (x - α)(x - β) + c = 0 are a and b, what is the relationship between α, β, a, and b?

If a and b are the roots of the quadratic equation x^2 + bx + c = 0, what is the value of a^2 + b^2 in terms of the coefficients?

If a quadratic equation has complex roots, what can be said about the discriminant?

What is the quadratic formula used to find?

If $α$ and $β$ are the roots of the equation $x^2 - ax + b = 0$, then what is the value of $α^2 + β^2$ in terms of $a$ and $b$?

If the roots of the equation $(x - α)(x - β) + c = 0$ are $a$ and $b$, then what is the value of $c$ in terms of $a$, $b$, $α$, and $β$?

If the roots of the equation $x^2 + ax + b = 0$ are $α$ and $β$, then what is the value of $αβ$ in terms of $a$ and $b$?

If α and β are the roots of the quadratic equation x² - Sx + P = 0, which of the following is true?

If we replace x by 1/x in the quadratic equation ax² + bx + c = 0, what is the resulting equation?

If the quadratic equation x² - Sx + P = 0 has roots α and β, what is the equation whose roots are 1/α and 1/β?

If α and β are the roots of the quadratic equation x² - Sx + P = 0, what is the value of (α + β)²?

If the quadratic equation ax² + bx + c = 0 has roots α and β, what is the equation whose roots are kα and kβ?

If α and β are the roots of the equation x² - Sx + P = 0, then what is the equation whose roots are -α and -β?

If α and β are the roots of the equation ax² + bx + c = 0, then what is the equation whose roots are 1/α and 1/β?

If α and β are the roots of the equation x² - Sx + P = 0, then what is the value of (α - β)² in terms of S and P?

If α and β are the roots of the equation ax² + bx + c = 0, then what is the equation whose roots are α² and β²?

If α and β are the roots of the equation x² - Sx + P = 0, then what is the value of α⁴ + β⁴ in terms of S and P?

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