Worked Examples: SAT Strategy
Example 1: Easy (Factoring and Roots)
Question: What is the positive solution to the given equation $x^2 - 8x - 20 = 0$?
SAT Strategy: When $a=1$, immediately look for two numbers that multiply to the constant ($c = -20$) and add to the middle coefficient ($b = -8$).
- The factors of $-20$ that add to $-8$ are $-10$ and $+2$.
- Rewrite the equation in factored form: $(x - 10)(x + 2) = 0$.
- Set each factor to zero to find the roots: $x = 10$ and $x = -2$.
- The question asks for the positive solution.
- Answer: $10$.
Example 2: Medium (Vertex Form & Minimums)
Question: The function $f$ is defined by $f(x) = 2x^2 - 12x + 22$. What is the minimum value of $f(x)$?
SAT Strategy: The minimum or maximum value of a quadratic function is the $y$-coordinate of its vertex. You can find the $x$-coordinate of the vertex using the formula $x = -\frac{b}{2a}$.
- Identify $a = 2$ and $b = -12$.
- Calculate the $x$-coordinate of the vertex: $x = -\frac{-12}{2(2)} = \frac{12}{4} = 3$.
- Substitute $x = 3$ back into the function to find the minimum value ($y$).
- $f(3) = 2(3)^2 - 12(3) + 22 = 2(9) - 36 + 22 = 18 - 36 + 22 = 4$.
- Answer: $4$.
Example 3: Hard (The Discriminant)
Question: The equation $x^2 - 14x + c = 0$ has exactly one real solution. If $c$ is a constant, what is the value of $c$?
SAT Strategy: A quadratic equation $ax^2 + bx + c = 0$ has exactly one real solution if and only if its discriminant ($\Delta = b^2 - 4ac$) is exactly equal to $0$. (If $>0$, two solutions. If $<0$, no real solutions).
- Identify the coefficients: $a = 1$, $b = -14$, and the constant is $c$.
- Set up the discriminant equation: $(-14)^2 - 4(1)(c) = 0$.
- Simplify: $196 - 4c = 0$.
- Add $4c$ to both sides: $196 = 4c$.
- Divide by 4: $c = 49$.
- Answer: $49$.
Practice Questions
Level 1: Easy (Foundational SAT)
- $(x - 4)(x + 9) = 0$. What is the positive solution to the given equation? (Grid-in)
- $x^2 - 25 = 0$. What is the positive solution to the given equation? (Grid-in)
- Which of the following expressions is equivalent to $x^2 + 7x + 10$?
- A) $(x + 2)(x + 5)$
- B) $(x - 2)(x - 5)$
- C) $(x + 1)(x + 10)$
- D) $(x + 3.5)^2 + 10$
- The function $f$ is defined by $f(x) = (x - 3)^2 + 5$. What are the coordinates of the vertex of the graph of $y = f(x)$ in the $xy$-plane?
- A) $(-3, 5)$
- B) $(3, -5)$
- C) $(3, 5)$
- D) $(9, 5)$
- $x^2 - 6x + 9 = 0$. How many distinct real solutions does the given equation have?
- A) Zero
- B) Exactly one
- C) Exactly two
- D) Infinitely many
Level 2: Medium (Standard SAT)
- $2x^2 - 16x + 30 = 0$. What is the sum of the solutions to the given equation?
- A) $-8$
- B) $8$
- C) $15$
- D) $30$
- The function $f$ is defined by $f(x) = x^2 - 10x + 24$. What is the minimum value of $f(x)$? (Grid-in)
- $x^2 + kx + 36 = 0$. In the given equation, $k$ is a positive constant. If the equation has exactly one real solution, what is the value of $k$? (Grid-in)
- $3x^2 + 12x - 15 = 0$. What is a positive solution to the given equation? (Grid-in)
- Which of the following expressions is equivalent to $2x^2 + 12x + 18$?
- A) $2(x + 3)^2$
- B) $2(x - 3)^2$
- C) $(2x + 3)^2$
- D) $(x + 6)^2 - 18$
Level 3: Hard (Advanced SAT)
- $ax^2 - 14x + 49 = 0$. In the given equation, $a$ is a constant. If the equation has no real solutions, which of the following could be the value of $a$?
- A) $0$
- B) $0.5$
- C) $1$
- D) $2$
- A parabola in the $xy$-plane intersects the $x$-axis at $(-2, 0)$ and $(6, 0)$ and passes through the point $(0, -12)$. What is the $y$-coordinate of the vertex of this parabola? (Grid-in)
- $x^2 - 8x + c = 0$. In the given equation, $c$ is a constant. If the solutions to the equation are $x = 4 + \sqrt{5}$ and $x = 4 - \sqrt{5}$, what is the value of $c$? (Grid-in)
- $y = x^2 - 4x + 3$
$y = 2x - 6$
The graphs of the given equations in the system intersect at exactly one point, $(x, y)$, in the $xy$-plane. What is the value of $x$? (Grid-in) - $x^2 + 6x = 40$. The given equation can be rewritten in the form $(x + 3)^2 = p$, where $p$ is a constant. What is the value of $p$? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- $2x^2 - bx + 18 = 0$. In the given equation, $b$ is a positive integer constant. If the equation has no real solutions, what is the greatest possible value of $b$? (Grid-in)
- The graph of $y = -2x^2 + 24x - c$ in the $xy$-plane has its vertex on the $x$-axis. What is the value of the constant $c$? (Grid-in)
- The quadratic function $g$ is defined by $g(x) = a(x - h)^2 + k$, where $a, h,$ and $k$ are constants. The vertex of the graph of $y = g(x)$ in the $xy$-plane is $(4, 10)$. If the graph passes through the point $(2, 2)$, what is the value of $g(6)$? (Grid-in)
- $3x^2 - 5x - 7 = 0$. Let $m$ and $n$ be the two solutions to the given equation. What is the value of $m + n$?
- A) $-\frac{7}{3}$
- B) $-\frac{5}{3}$
- C) $\frac{5}{3}$
- D) $\frac{7}{3}$
- The graph of $y = (x - 2)(x + 8)$ is a parabola in the $xy$-plane. A horizontal line defined by the equation $y = c$ intersects the parabola at exactly one point. What is the value of $c$? (Grid-in)
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