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College Algebra: Quadratic Functions and Factoring

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Termini in questo insieme (19)

  • What are the zeros of a function?
    The zeros of a function are the values of the variable that make the function equal to zero.
  • How do you find the zeros of a quadratic function?
    Set the quadratic equation equal to zero and solve for the variable using factoring, completing the square, or the quadratic formula.
  • What is the formula to determine profit using a quadratic equation?
    Profit P(x) can be modeled as a quadratic function, and zeros of P(x) represent break-even points where \(P(x)=0\).
  • How do you find the second solution of an imaginary number given one solution?
    If one solution is \(a+bi\), the second solution is its conjugate \(a-bi\).
  • How do you factor a quadratic equation with square roots?
    Express the quadratic in the form \((\sqrt{a}x + \sqrt{b})^2\) or use difference of squares if applicable.
  • How do you factor a quadratic equation with fractions?
    Multiply the entire equation by the least common denominator to clear fractions, then factor normally.
  • How do you factor quadratic equations with letters other than x and y?
    Use the same factoring methods, treating the letters as variables. For example, factor \(az^2 + bz + c\) using the formula \(az^2 + bz + c = (mz + n)(pz + q)\).
  • What is the quadratic formula?
    The quadratic formula is \(x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), used to find the roots of \(ax^2 + bx + c = 0\).
  • How do you factor a quadratic using the difference of squares?
    If the quadratic is in the form \(a^2 - b^2\), factor as \((a - b)(a + b)\).
  • What is the conjugate of the imaginary number \(3 + 4i\)?
    The conjugate is \(3 - 4i\).
  • How do you find zeros of a quadratic function by factoring?
    Set the quadratic equal to zero, factor it, then set each factor equal to zero and solve.
  • What is the first step in factoring a quadratic with fractions?
    Clear fractions by multiplying both sides by the least common denominator.
  • How do you factor \(x^2 - 9\)?
    Use difference of squares: \((x - 3)(x + 3)\).
  • How do you factor \(4x^2 - 25y^2\)?
    Use difference of squares: \((2x - 5y)(2x + 5y)\).
  • What is the profit function if revenue is \(R(x) = 50x\) and cost is \(C(x) = 10x + 200\)?
    Profit is \(P(x) = R(x) - C(x) = 50x - (10x + 200) = 40x - 200\).
  • How do you find the zeros of the profit function \(P(x) = 40x - 200\)?
    Set \(40x - 200 = 0\) and solve for \(x\): \(x = 5\).
  • How do you factor \(x^2 + 5x + 6\)?
    Find two numbers that multiply to 6 and add to 5: 2 and 3. Factor as \((x + 2)(x + 3)\).
  • How do you factor \(2z^2 + 7z + 3\) using the AC method?
    Multiply a and c: 2*3=6. Find factors of 6 that add to 7: 6 and 1. Rewrite as \(2z^2 + 6z + z + 3\) and factor by grouping.
  • What is the formula for factoring quadratic equations with letters other than x?
    For \(az^2 + bz + c\), factor as \((mz + n)(pz + q)\) where \(m p = a\), \(n q = c\), and \(m q + n p = b\).