Quadratics & Polynomials - ACT Math

Factor: $x^2 - 9$
Answer: $(x - 3)(x + 3)$ To factor $x^2 - 9$ , we recognize this as a difference of squares pattern $a^2 - b^2 = (a + b)(a - b)$ . Here we have $x^2 - 3^2$ , so the factorization is $(x + 3)(x - 3)$ . We can verify: $(x + 3)(x - 3) = x^2 - 3x + 3x - 9 = x^2 - 9$ .
A profit model uses the expression $x^2-9x+20$ . Which is the correct factorization of $x^2-9x+20$ ?
Answer: $\left(x-4\right)\left(x-5\right)$ To factor x² - 9x + 20, find two numbers that multiply to 20 and add to -9. Since (-4)(-5) = 20 and (-4) + (-5) = -9, the factorization is (x - 4)(x - 5). We can verify: (x - 4)(x - 5) = x² - 5x - 4x + 20 = x² - 9x + 20. Choice B would give a positive middle coefficient.
Which is the correct factorization of $x^2 - 4x - 5$ ?
Answer: $(x - 5)(x + 1)$ To factor $x^2 - 4x - 5$ , we need two numbers that multiply to -5 and add to -4. The numbers -5 and 1 work: (-5)(1) = -5 and (-5) + 1 = -4. Therefore, the factorization is $(x - 5)(x + 1)$ . We can verify: $(x - 5)(x + 1) = x^2 + x - 5x - 5 = x^2 - 4x - 5$ .
Which is the correct factorization of $x^2 - 5x + 6$ ?
Answer: $(x - 2)(x - 3)$ To factor x² - 5x + 6, we need two numbers that multiply to 6 and add to -5. The numbers are -2 and -3 since (-2)(-3) = 6 and -2 + (-3) = -5. Therefore, x² - 5x + 6 = (x - 2)(x - 3). Choice C has factors that multiply to 6 but add to -7.
A quadratic equation for a design constraint is $x^2-3x-10=0$ . What are the solutions to $x^2-3x-10=0$ ?
Answer: $x=5$ and $x=-2$ To solve x² - 3x - 10 = 0, factor by finding two numbers that multiply to -10 and add to -3. Since (2)(-5) = -10 and 2 + (-5) = -3, we get (x - 5)(x + 2) = 0. Setting each factor equal to zero gives x = 5 and x = -2. Choice B has the wrong signs for both solutions.
A ball's height is modeled by $h(t)=t^2-5t+6$ . To find when it hits the ground, solve $t^2-5t+6=0$ .
Answer: $t=2$ and $t=3$ To solve $t^2 - 5t + 6 = 0$ , factor by finding two numbers that multiply to 6 and add to -5. Since $(-2)(-3) = 6$ and $(-2) + (-3) = -5$ , we get $(t - 2)(t - 3) = 0$ . Setting each factor equal to zero gives $t = 2$ and $t = 3$ . Choice A uses the quadratic formula unnecessarily, and choices C and D give incorrect values.
Solve the equation $x^2 - 8x + 15 = 0$ by factoring.
Answer: $(3, 5)$ To solve $x^2 - 8x + 15 = 0$ by factoring, we need two numbers that multiply to 15 and add to -8. The numbers -3 and -5 satisfy this: (-3)(-5) = 15 and (-3) + (-5) = -8. So we factor as $(x - 3)(x - 5) = 0$ , giving solutions $x = 3$ and $x = 5$ , which we write as $(3, 5)$ .
Solve: $x^2 + 4x - 21 = 0$
Answer: $(-7, 3)$ To solve $x^2 + 4x - 21 = 0$ by factoring, we need two numbers that multiply to -21 and add to 4. The numbers 7 and -3 work: (7)(-3) = -21 and 7 + (-3) = 4. Factoring gives $(x + 7)(x - 3) = 0$ , so $x = -7$ or $x = 3$ , making the solutions $(-7, 3)$ .
What are the solutions to the quadratic equation $x^2 - 3x - 10 = 0$ ?
Answer: 5 and -2 To solve x² - 3x - 10 = 0, we need two numbers that multiply to -10 and add to -3. The numbers are -5 and 2 since (-5)(2) = -10 and -5 + 2 = -3. Factoring gives us (x - 5)(x + 2) = 0, so x = 5 or x = -2. The solutions are 5 and -2.
Factor: $x^2 + 3x - 10$
Answer: $(x + 5)(x - 2)$ To factor $x^2 + 3x - 10$ , we need two numbers that multiply to -10 and add to 3. The numbers are 5 and -2 since $(5)(-2) = -10$ and $5 + (-2) = 3$ . Therefore, $x^2 + 3x - 10 = (x + 5)(x - 2)$ . Choice B has the wrong signs for the factors.
A polynomial used in a calculation is $x^2-6x+9$ . Factor: $x^2-6x+9$ .
Answer: $\left(x-3\right)^2$ The expression x² - 6x + 9 is a perfect square trinomial of the form a² - 2ab + b² where a = x and b = 3. Since (-6x) = 2(x)(-3) and 9 = (-3)², we get x² - 6x + 9 = (x - 3)². We can verify: (x - 3)² = x² - 6x + 9. Choice A would be a difference of squares.
Given $f(x)=x^2-2x+1$ , what is the value of $f(3)$ ?
Answer: $4$ To find f(3) where f(x) = x² - 2x + 1, substitute x = 3 into the function. Calculate: f(3) = (3)² - 2(3) + 1 = 9 - 6 + 1 = 4. Notice that f(x) = x² - 2x + 1 = (x - 1)², so f(3) = (3 - 1)² = 4.
Factor: $x^2 - 6x + 9$
Answer: $(x - 3)^2$ To factor $x^2 - 6x + 9$ , we recognize this as a perfect square trinomial. We need two numbers that multiply to 9 and add to -6: (-3)(-3) = 9 and (-3) + (-3) = -6. This gives us $(x - 3)(x - 3) = (x - 3)^2$ . Perfect square trinomials follow the pattern $a^2 - 2ab + b^2 = (a - b)^2$ .
Factor: $x^2 + 7x + 10$
Answer: $(x + 5)(x + 2)$ To factor $x^2 + 7x + 10$ , we need two numbers that multiply to 10 and add to 7. The numbers 5 and 2 satisfy this condition: (5)(2) = 10 and 5 + 2 = 7. Therefore, the factorization is $(x + 5)(x + 2)$ . We can verify by expanding: $(x + 5)(x + 2) = x^2 + 2x + 5x + 10 = x^2 + 7x + 10$ .
Factor: $x^2 + 6x + 9$
Answer: $(x + 3)^2$ To factor $x^2 + 6x + 9$ , we recognize this as a perfect square trinomial. We need two numbers that multiply to 9 and add to 6: $(3)(3) = 9$ and $3 + 3 = 6$ . This gives us $(x + 3)(x + 3) = (x + 3)^2$ . Perfect square trinomials follow the pattern $a^2 + 2ab + b^2 = (a + b)^2$ .
A revenue expression is $x^2+x-12$ . Factor: $x^2+x-12$ .
Answer: $\left(x-3\right)\left(x+4\right)$ To factor x² + x - 12, find two numbers that multiply to -12 and add to 1. Since (4)(-3) = -12 and 4 + (-3) = 1, the factorization is (x + 4)(x - 3). Rearranging gives (x - 3)(x + 4). We can verify: (x - 3)(x + 4) = x² + 4x - 3x - 12 = x² + x - 12.
Which is the correct factorization of $x^2 - 6x + 8$ ?
Answer: $(x - 4)(x - 2)$ To factor x² - 6x + 8, we need two numbers that multiply to 8 and add to -6. The numbers are -4 and -2 since (-4)(-2) = 8 and -4 + (-2) = -6. Therefore, x² - 6x + 8 = (x - 4)(x - 2). Choice C represents a perfect square that would give x² - 6x + 9.
What is the value of $f(-1)$ for $f(x) = x^2 + 2x - 3$ ?
Answer: -4 To find f(-1) for f(x) = x² + 2x - 3, substitute x = -1 into the function. f(-1) = (-1)² + 2(-1) - 3 = 1 - 2 - 3 = -4. Therefore, f(-1) = -4.
A rectangular garden has an area represented by $x^2+7x+12$ square feet, where $x$ is the width added in a redesign. Factor: $x^2+7x+12$ .
Answer: $\left(x+3\right)\left(x+4\right)$ To factor $x^2 + 7x + 12$ , find two numbers that multiply to 12 and add to 7. Since $(3)(4) = 12$ and $3 + 4 = 7$ , the factorization is $(x + 3)(x + 4)$ . We can verify: $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ . Choice C $(x + 2)(x + 6)$ gives wrong factors since $2 + 6 = 8$ , not 7.
Which is the correct factorization of $x^2 - 4x - 12$ ?
Answer: $(x - 6)(x + 2)$ To factor x² - 4x - 12, we need two numbers that multiply to -12 and add to -4. The numbers are -6 and 2 since (-6)(2) = -12 and -6 + 2 = -4. Therefore, x² - 4x - 12 = (x - 6)(x + 2). Choice D has factors that give the wrong middle term.

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