SAT Math: Passport to Advanced Math

All 14 cards

1. 1

   If the quadratic equation 3x^2 - 6x + c = 0 has exactly one real solution, what is the value of c?

   3. For a quadratic equation ax^2 + bx + c = 0 to have exactly one real solution, its discriminant must equal zero (b^2 - 4ac = 0). Substituting the values: (-6)^2 - 4(3)(c) = 0 -> 36 - 12c = 0 -> c = 3.
2. 2

   What is the sum of the solutions to the quadratic equation 2x^2 - 8x + 5 = 0?

   4. The sum of the solutions to any quadratic equation in the form ax^2 + bx + c = 0 is given by -b/a. Substituting the coefficients: -(-8)/2 = 4.
3. 3

   What is the product of the solutions to the quadratic equation 3x^2 + 7x - 12 = 0?

   -4. The product of the solutions to any quadratic equation in the form ax^2 + bx + c = 0 is given by c/a. Substituting the coefficients: -12/3 = -4.

4. 4

   For the quadratic function f(x) = -2(x - 3)^2 + 8, what is the maximum value of the function?

   8. The vertex of a quadratic in vertex form, f(x) = a(x - h)^2 + k, is (h, k). Here, the vertex is (3, 8). Since the coefficient 'a' is negative (-2), the parabola opens downward, meaning the y-coordinate of the vertex (8) is the maximum value.
5. 5

   If x > 0, rewrite the radical expression sqrt[3]{x^5} (the cube root of x to the fifth power) using a rational exponent.

   x^(5/3). The radical-to-exponent rule states that sqrt[n]{x^m} = x^(m/n). Here, the power m is 5 and the root index n is 3.

6. 6

   Simplify the expression ((x³⁾4) / (x^-2) into a single power of x.

   x^14. First apply the power-to-a-power rule: (x³⁾4 = x^12. Then apply the quotient rule: (x^12) / (x^-2) = x^(12 - (-2)) = x^14.

7. 7

   If a polynomial P(x) is divided by x - 4 and the remainder is 7, what is the value of P(4)?

   7. According to the Remainder Theorem, if a polynomial P(x) is divided by x - c, the remainder is equal to P(c). Here, c = 4, so P(4) = 7.
8. 8

   Why must you check for extraneous solutions when solving equations containing radicals or variables in the denominator?

   Certain operations (such as squaring both sides of an equation or multiplying by a variable) can introduce solutions that are mathematically valid for the transformed equation but fail to satisfy the constraints of the original equation.

9. 9

   In the exponential model P(t) = 250(0.88)^t, what does the base 0.88 represent?

   The decay factor. It indicates that the quantity P(t) decreases by 12% (1 - 0.88 = 0.12) during each unit of time t.

10. 10

    If a polynomial f(x) has factors (x - 1), (x + 2), and (x - 5), what are the x-coordinates of all x-intercepts of the graph of f(x)?

    1, -2, and 5. The x-intercepts occur where f(x) = 0. Setting each factor to zero (x - 1 = 0, x + 2 = 0, and x - 5 = 0) gives the roots x = 1, x = -2, and x = 5.

11. 11

    To find the coordinates of the intersection points of the system y = x^2 - 4x + 3 and y = 2x - 2, what single quadratic equation must you solve first?

    x^2 - 6x + 5 = 0. Set the two equations equal to each other: x^2 - 4x + 3 = 2x - 2. Subtract 2x and add 2 to both sides to write the equation in standard form.

12. 12

    What is the x-coordinate of the vertex of the parabola defined by the equation y = 3x^2 + 12x - 5?

    -2. For any parabola in standard form y = ax^2 + bx + c, the x-coordinate of the vertex is given by x = -b/(2a). Substituting the values: -12 / (2 * 3) = -2.

13. 13

    Simplify the numerical expression (1 / 64)^(-1/3).

    4. First, resolve the negative exponent by taking the reciprocal of the base: (1/64)^(-1/3) = (64)^(1/3). Next, resolve the fractional exponent by taking the cube root: 64^(1/3) = 4.
14. 14

    If a polynomial A(x) is divided by B(x), it can be written as A(x)/B(x) = Q(x) + R(x)/B(x), where Q(x) is the quotient and R(x) is the ___.

    remainder. R(x) is the remainder polynomial, which must have a degree strictly less than the degree of the divisor B(x).