Quadratic Functions and Factoring Practice Quiz (16 questions)

1. For a quadratic in standard form, $y=ax^2+bx+c$, which statement correctly describes the roles of $a$ and $c$?

A. $c$ gives the zeros, while $a$ determines the vertex.

B. $a$ determines opening direction and shape, while $c$ gives the y-intercept.

C. $a$ gives the vertex, while $c$ determines the axis of symmetry.

D. $c$ determines opening direction and shape, while $a$ gives the y-intercept.

Answer: (B)

Explanation

The leading coefficient $a$ affects the parabola’s opening direction and shape, and the constant term $c$ is its y-intercept. Confusing their roles would incorrectly make the constant term control the opening.

2. A quadratic is written as $y=-2(x-3)^2+5$. What is its vertex?

A. $(-3,5)$

B. $(-2,5)$

C. $(3,-5)$

D. $(3,5)$

Answer: (D)

Explanation

In vertex form $y=a(x-h)^2+k$, the vertex is $(h,k)$, so here it is $(3,5)$. The negative sign outside the squared expression affects the parabola’s opening, not the vertex coordinates.

3. A quadratic is expressed as $y=(x-4)(x+2)$. What do the values that make either factor zero represent on its graph?

A. The turning point of the parabola

B. The axis of symmetry of the parabola

C. The y-intercept of the parabola

D. The x-intercepts of the parabola

Answer: (D)

Explanation

Values that make a factor zero are the quadratic’s zeros, which correspond to its x-intercepts. The turning point is the vertex, not either zero.

4. Which procedure correctly expands the squared binomial $(x+3)^2$?

A. Write $(x+3)(x-3)$, multiply the factors, and simplify.

B. Write $(x+3)(x+3)$, multiply the factors, and simplify.

C. Write $x^2+3^2$, then combine the squared terms.

D. Write $x(x+3)+3$, then collect like terms.

Answer: (B)

Explanation

A squared binomial means two identical factors, so $(x+3)^2=(x+3)(x+3)$; multiplying and simplifying gives the expansion. Using opposite-sign factors instead would expand a difference of squares, not the given square.

5. When converting $y=2(x-1)(x+4)$ to standard form, which sequence of operations is appropriate?

A. Expand the brackets, distribute the outside coefficient, and collect like terms.

B. Collect terms inside each factor, square the factors, and subtract the coefficient.

C. Distribute the coefficient to each factor, combine the factors, and leave the result factored.

D. Find the zeros, use them as coefficients, and then add the outside coefficient.

Answer: (A)

Explanation

To convert from factored form to standard form, expand the brackets, distribute any outside coefficient, and collect like terms. Finding the zeros does not perform the expansion into standard form.

6. What determines the greatest common factor of several polynomial terms?

A. The lowest shared coefficient and the greatest exponent of each shared variable

B. The greatest shared coefficient and the lowest exponent of each shared variable

C. The greatest coefficient and greatest exponent appearing in any term

D. The sum of the coefficients and the variables’ average exponents

Answer: (B)

Explanation

The greatest common factor uses the greatest coefficient shared by the terms and the lowest exponent of every variable shared among them. Choosing the greatest exponent could produce a factor that does not divide every term.

7. To factor $x^2+7x+12$ when the leading coefficient is 1, which pair of constants should be used in the binomials?

A. 2 and 6, because their product is 12 and their difference is 7

B. 1 and 12, because their product is 12 and their difference is 7

C. -3 and -4, because their product is 12 and their sum is 7

D. 3 and 4, because their product is 12 and their sum is 7

Answer: (D)

Explanation

For a monic trinomial, choose a factor pair of the constant term whose sum equals the middle coefficient; 3 and 4 multiply to 12 and add to 7. The pair 2 and 6 has the correct product but does not have the required sum or difference.

8. Which method is used to factor a trinomial $ax^2+bx+c$ when $a\ne1$?

A. Find two factors of $c$ whose sum is $b$, then place them directly in binomials with leading coefficient $a$.

B. Multiply $a$ and $c$, split the middle term using coefficients with product $ac$ and sum $b$, then factor by grouping.

C. Divide $b$ by $a$, split the constant using terms with product $b$ and sum $c$, then factor by grouping.

D. Complete the square using $a$ and $c$, then factor the resulting difference of squares.

Answer: (B)

Explanation

The method uses the product $ac$ to split the middle term into coefficients that add to $b$, then groups the terms to extract a common binomial. Choosing factors of $c$ whose sum is $b$ is the simpler strategy for a trinomial whose leading coefficient is 1.

9. Which expression qualifies as a difference of two squares?

A. A trinomial whose middle term is twice two square roots

B. A binomial formed by adding two perfect-square terms

C. A trinomial with positive square terms and a middle term

D. A binomial formed by subtracting two perfect-square terms

Answer: (D)

Explanation

A difference of two squares has two terms, subtracts one perfect square from another, and can be factored using their square roots. A trinomial with a middle term describes a different factoring pattern.

10. How does the expression $A^2-B^2$ factor?

A. $(A^2+B)(A^2-B)$

B. $(A-B)^2$

C. $(A+B)^2$

D. $(A+B)(A-B)$

Answer: (D)

Explanation

The difference of squares factors as the sum of the square roots multiplied by their difference. Squaring a binomial instead produces a trinomial, not the original two-term expression.

11. A trinomial has positive perfect-square first and last terms, and its middle term is twice the product of their square roots. What factoring pattern applies?

A. It factors as a product of three linear factors, with the middle term determining their signs.

B. It factors into two binomials with opposite signs between corresponding terms.

C. It factors into the sum and difference of the first and last square roots.

D. It factors as a repeated binomial squared, with the binomial sign matching the middle term.

Answer: (D)

Explanation

These features identify a perfect-square trinomial, which factors as a binomial squared with the sign of the middle term. A difference of squares has two terms and factors into opposite-sign binomials.

12. Which sequence correctly solves a quadratic equation by factoring?

A. Set each term equal to zero, solve those equations, and then factor the quadratic.

B. Move all terms to one side so the equation equals zero, factor, set each factor to zero, and solve.

C. Factor both sides, combine like factors, and set the original variable equal to zero.

D. Move the variable terms to one side, take square roots, and factor any remaining constants.

Answer: (B)

Explanation

Factoring to solve requires writing the equation as zero, factoring, and using the zero-product property on the factors. Taking square roots is a different approach and does not replace the required factoring steps.

13. What are the solutions of $x^2-5x-50=0$ after factoring?

A. $x=-10$ and $x=5$

B. $x=-10$ and $x=-5$

C. $x=10$ and $x=5$

D. $x=10$ and $x=-5$

Answer: (A)

Explanation

The equation factors as $(x+10)(x-5)=0$, giving roots $x=-10$ and $x=5$. Reversing both signs would not satisfy the two factor equations.

14. A quadratic graph has a negative leading coefficient. Which way does its parabola open?

A. To the right

B. To the left

C. Upward

D. Downward

Answer: (D)

Explanation

A negative leading coefficient makes a quadratic parabola open downward. An upward-opening parabola instead has a positive leading coefficient.

15. What happens at the vertex of a quadratic function's graph?

A. The graph reaches the point where its leading coefficient becomes zero.

B. The graph changes from increasing to decreasing, or from decreasing to increasing.

C. The graph crosses the x-axis and changes the sign of its y-values.

D. The graph changes from opening upward to opening downward.

Answer: (B)

Explanation

The vertex is where a quadratic graph switches between increasing and decreasing. Crossing the x-axis concerns zeros and does not define the vertex.

16. For a quadratic function, what do its positive intervals describe?

A. The x-values where the function's y-values are at or above the x-axis, including zeros.

B. The x-values where the function's y-values increase as x increases.

C. The x-values where the function's y-values are above the x-axis, with strict inequalities.

D. The x-values where the function's y-values are below the x-axis, with strict inequalities.

Answer: (C)

Explanation

Positive intervals identify the x-values where the quadratic's output is above the x-axis, and the inequalities are strict. Intervals where outputs are below the axis are negative intervals instead.