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Precalculus: Equations, Inequalities, and Problem Solving

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Terms in this set (21)

  • How do you solve a linear equation?
    Isolate the variable by performing inverse operations to both sides until the variable stands alone.
  • What is the first step to solve equations that lead to linear equations?
    Simplify the equation by expanding, combining like terms, or factoring to reduce it to a linear form.
  • How do you solve quadratic equations by factoring?
    Rewrite the quadratic in factored form, set each factor equal to zero, then solve for the variable.
  • What does completing the square involve?
    Rewrite a quadratic equation in the form \((x+p)^2=q\) by adding a constant to both sides.
  • How do you solve a quadratic equation by completing the square?
    Convert the quadratic to a perfect square trinomial, then take the square root of both sides and solve for the variable.
  • What is the quadratic formula used for?
    To find the roots of any quadratic equation \(ax^2+bx+c=0\) using \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
  • How do you solve quadratic equations using the square root property?
    Isolate the squared term, then take the square root of both sides, remembering to include ± for two solutions.
  • How are quadratic equations solved in the complex number system?
    Use the quadratic formula; if the discriminant is negative, express solutions with imaginary unit \(i\).
  • What is a radical equation quadratic in form?
    An equation where a variable is under a root and can be rewritten as a quadratic by substitution.
  • How do you solve radical equations by factoring?
    Isolate the radical, square both sides to eliminate it, then factor the resulting quadratic and solve.
  • What is interval notation used for in inequalities?
    To express the solution set of inequalities as intervals on the number line.
  • What are key properties of inequalities?
    You can add, subtract, multiply, or divide both sides by the same positive number without changing the inequality direction; multiplying/dividing by a negative reverses it.
  • How do you solve combined inequalities?
    Solve each inequality separately and find the intersection (and) or union (or) of their solution sets.
  • How do you solve equations involving absolute value?
    Set up two separate equations: one with the expression inside equal to the positive value, one equal to the negative value.
  • How do you solve inequalities involving absolute value?
    Rewrite as two inequalities without absolute value, considering cases for less than or greater than.
  • What is the distance formula?
    Distance between points \((x_1,y_1)\text{ and }(x_2,y_2)\) is \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).
  • What is the midpoint formula?
    Midpoint between points \((x_1,y_1)\text{ and }(x_2,y_2)\) is \(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\).
  • How do you translate verbal descriptions into mathematical expressions?
    Identify variables, keywords, and relationships, then write corresponding algebraic expressions or equations.
  • How do you solve interest problems in precalculus?
    Use formulas for simple or compound interest and set up equations based on given conditions.
  • How do you solve mixture problems?
    Set up equations based on the amounts and concentrations of components being mixed.
  • How are distance and midpoint formulas used in problem solving?
    Apply them to solve problems involving geometry, interest, or mixtures that relate to points on a coordinate plane.