Tayarri LCAT
LCAT Math, Advanced Math

Quadratics for the LCAT: lesson and practice

LCAT quadratic questions test four things: solving (factor, take square roots, or use the quadratic formula), the vertex (the minimum or maximum point, at x = −b/(2a)), the sum and product of solutions (−b/a and c/a), and the number of solutions (from the discriminant b² − 4ac).

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No calculator on the LCAT. LUMS’s official guidelines say calculators are not allowed, so every practice question here can be solved by hand.

Solving

FormBest methodExample
x² + bx + c = 0Factorx² − 5x − 14 = (x − 7)(x + 2) → 7, −2
(x − h)² = kSquare root, both signs(x − 3)² = 16 → x = 7 or −1
Anything elseQuadratic formulax = (−b ± √(b² − 4ac)) / 2a

The vertex

In y = a(x − h)² + k, the vertex is (h, k). In y = ax² + bx + c, it is at x = −b/(2a); substitute to get the minimum or maximum value.

Two shortcuts

Asked forFormula
Sum of solutions−b/a
Product of solutionsc/a
Number of real solutionsb² − 4ac: positive → 2, zero → 1, negative → 0

Common mistakes

Wrong signs from factors(x − 7) gives x = 7, not −7.
Only the positive square root(x − 3)² = 16 has two solutions.
Squaring a negative(−3)² = 9.
Giving the x-value instead of the minimum valueThe minimum value is the y-value of the vertex.

Sample questions with answers

1. What are the solutions to x² − 5x − 14 = 0?
  1. x = −7 and x = 2
  2. x = 7 and x = −2 (correct)
  3. x = 14 and x = −1
  4. x = 5 and x = −14

(x − 7)(x + 2) = 0, so x = 7 or x = −2.

2. What is the vertex of y = 2(x + 3)² − 5?
  1. (3, −5)
  2. (−3, −5) (correct)
  3. (−3, 5)
  4. (−5, −3)

x + 3 = x − (−3), so h = −3 and k = −5.

3. How many real solutions does x² + 4x + 5 = 0 have?
  1. Zero (correct)
  2. Exactly one
  3. Exactly two
  4. Infinitely many

16 − 20 = −4 is negative, so there are no real solutions.

Frequently asked questions

How do I find the minimum value of a quadratic?

Find the vertex x-value with −b/(2a), then substitute it into the function. The result is the minimum (if a > 0).

What is the discriminant used for?

b² − 4ac tells you how many real solutions a quadratic has: positive means two, zero means one, negative means none.

Is quadratics tested on the LCAT?

Yes. It falls under Advanced Math, one of the content domains LUMS lists for the LCAT in its official sample questions.

Can I practice this without a calculator?

Yes. Every practice question here can be solved by hand, because calculators are not allowed on the LCAT.

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