Flashcard Set

SAT Math: Passport to Advanced Math

14 cards

Exam-grade practice on quadratics, exponents, polynomials, and nonlinear functions for the digital SAT.

1
Term
If the quadratic equation 3x^2 - 6x + c = 0 has exactly one real solution, what is the value of c?
Definition
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
Term
What is the sum of the solutions to the quadratic equation 2x^2 - 8x + 5 = 0?
Definition
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
Term
What is the product of the solutions to the quadratic equation 3x^2 + 7x - 12 = 0?
Definition
-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
Term
For the quadratic function f(x) = -2(x - 3)^2 + 8, what is the maximum value of the function?
Definition
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
Term
If x > 0, rewrite the radical expression sqrt[3]{x^5} (the cube root of x to the fifth power) using a rational exponent.
Definition
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
Term
Simplify the expression ((x^3)^4) / (x^-2) into a single power of x.
Definition
x^14. First apply the power-to-a-power rule: (x^3)^4 = x^12. Then apply the quotient rule: (x^12) / (x^-2) = x^(12 - (-2)) = x^14.
7
Term
If a polynomial P(x) is divided by x - 4 and the remainder is 7, what is the value of P(4)?
Definition
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
Term
Why must you check for extraneous solutions when solving equations containing radicals or variables in the denominator?
Definition
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
Term
In the exponential model P(t) = 250(0.88)^t, what does the base 0.88 represent?
Definition
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
Term
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)?
Definition
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
Term
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?
Definition
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
Term
What is the x-coordinate of the vertex of the parabola defined by the equation y = 3x^2 + 12x - 5?
Definition
-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
Term
Simplify the numerical expression (1 / 64)^(-1/3).
Definition
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
Term
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 ___.
Definition
remainder. R(x) is the remainder polynomial, which must have a degree strictly less than the degree of the divisor B(x).