Test yourself on Quadratics with 6 original SAT practice questions. Pick an answer to see instant feedback and a full explanation.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of SAT.
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Answer the questions below — you get instant feedback and a full explanation for each.
1. What are the solutions to the equation x² - 5x + 6 = 0?
Explanation. Factor: x² - 5x + 6 = (x - 2)(x - 3) = 0. Setting each factor to zero gives x = 2 and x = 3. Check: 2 + 3 = 5 (the sum equals the negative of the middle coefficient) and 2·3 = 6 (the product equals the constant). Negative options would give a positive middle term.
2. The quadratic function f(x) = x² - 6x + 11 has its vertex at which point?
Explanation. The x-coordinate of the vertex is x = -b/(2a) = -(-6)/(2·1) = 3. Then f(3) = 9 - 18 + 11 = 2, so the vertex is (3, 2). Equivalently, completing the square gives (x-3)² + 2.
3. For what value of c does the equation x² + 8x + c = 0 have exactly one real solution?
Explanation. A quadratic has exactly one real solution when the discriminant b² - 4ac = 0. Here 8² - 4(1)(c) = 64 - 4c = 0, so c = 16. This makes the equation (x+4)² = 0, with the double root x = -4.
4. If the solutions to 2x² + bx - 6 = 0 are x = -3 and x = 1, what is the value of b?
Explanation. For ax² + bx + c, the sum of roots is -b/a. The sum of roots is -3 + 1 = -2, so -b/2 = -2, giving b = 4. Check product: (-3)(1) = -3 = c/a = -6/2 = -3. ✓
5. The expression x² + 10x + 7 can be written in the form (x + a)² + b. What is the value of b?
Explanation. Complete the square: take half of 10 to get 5, and 5² = 25. So x² + 10x + 7 = (x² + 10x + 25) - 25 + 7 = (x + 5)² - 18. Thus a = 5 and b = -18.
6. A ball's height in feet is given by h(t) = -16t² + 32t + 5, where t is time in seconds. What is the maximum height of the ball?
Explanation. Maximum height occurs at the vertex: t = -b/(2a) = -32/(2·(-16)) = 1 second. Then h(1) = -16(1) + 32(1) + 5 = -16 + 32 + 5 = 21 feet. The value 5 is only the starting height at t = 0.
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FAQ
What are the main methods to solve quadratic equations on the SAT?
Factoring (fastest when roots are integers), using the quadratic formula x = (-b ± √(b²-4ac))/(2a) for any quadratic, and completing the square (useful for vertex form). Recognizing perfect-square trinomials and difference of squares speeds things up.
How do I find the vertex of a parabola quickly?
For f(x) = ax² + bx + c, the vertex x-coordinate is x = -b/(2a). Plug that back into the function to get the y-coordinate. The vertex gives the maximum (if a < 0) or minimum (if a > 0) value, which is commonly tested in word problems.
What does the discriminant tell me?
The discriminant b² - 4ac determines the number of real solutions: positive means two real solutions, zero means exactly one (a repeated root), and negative means no real solutions. SAT problems often ask you to find a coefficient that produces one or no real solutions.