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Quadratic equations — ACT practice questions

Quadratic equations on the ACT Mathematics test cover finding zeros, rewriting equivalent forms, and using a known root to find a missing coefficient. A student must expand vertex form, complete the square or factor, apply the quadratic formula, and use root-coefficient relationships. Items typically ask which choice gives both zeros, which equation is equivalent after rearranging, or the value of a parameter that makes a given solution work. Common traps include listing only one zero, dropping a sign when moving terms, and treating a squared binomial as already solved. Answer choices usually list pairs of roots, rewritten equations, or one coefficient.

  • Section: Mathematics 45 questions on the paper
  • 43 practice questions in the app

Sample questions

Question 1

Which of the following gives both zeros of f(x)=2(x−3)2−18f(x)=2(x-3)^2-18?

  1. A-6 and 0
  2. B0 and 6
  3. C3 and 9
  4. D-3 and 3
Show answer

Answer: B — 0 and 6

Setting f(x)=0f(x)=0 gives 2((x−3)2−9)=02\left((x-3)^2-9\right)=0, so 2x(x−6)=02x(x-6)=0. Therefore the zeros are 0 and 6.

Question 2

Which of the following is equivalent to x2+10x=11x^2+10x=11?

  1. A(x+5)2=36(x+5)^2=36
  2. B(x+5)2=11(x+5)^2=11
  3. C(x+10)2=111(x+10)^2=111
  4. D(x−5)2=36(x-5)^2=36
Show answer

Answer: A — (x+5)2=36(x+5)^2=36

Adding 25 to both sides gives x2+10x+25=36x^2+10x+25=36. The left side factors as (x+5)2(x+5)^2, so (x+5)2=36(x+5)^2=36.

Question 3

One solution of the equation x2−bx+18=0x^2-bx+18=0 is x=3x=3. What is the value of bb?

  1. A5
  2. B6
  3. C9
  4. D15
Show answer

Answer: C — 9

Substitute x=3x=3: 9−3b+18=0⇒27=3b⇒b=99-3b+18=0\Rightarrow 27=3b\Rightarrow b=9.

Practice 43 Quadratic equations questions in the app

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