This question tests your ability to factor quadratic expressions and use the factored form to identify zeros (x-intercepts) of the function—essential for graphing and solving quadratic equations. Factoring a quadratic into form f(x) = a(x - r)(x - s) reveals the zeros immediately: set the factored expression equal to zero and use the zero product property (if a product equals zero, at least one factor must equal zero). Setting (x - r) = 0 gives x = r, and setting (x - s) = 0 gives x = s, so the zeros are r and s—these are the x-intercepts of the parabola at points (r, 0) and (s, 0) where the graph crosses the x-axis. For f(x) = x² + 2x - 8, numbers multiply to -8, add to 2: 4 and -2, so (x + 4)(x - 2); zeros x = -4 and x = 2, x-intercepts (-4, 0) and (2, 0)—one positive, one negative! Choice A correctly factors and finds the zeros. Choice B swaps signs, expanding to x² - 2x - 8—close but middle term wrong; always FOIL to check! For mixed signs, choose pairs with opposite signs; this helps sketch parabolas quickly—keep going, you're doing great!

This question tests your ability to factor quadratic expressions and use the factored form to identify zeros (x-intercepts) of the function—essential for graphing and solving quadratic equations; here, zeros are break-even points. Factoring a quadratic into form f(x) = a(x - r)(x - s) reveals the zeros immediately: set the factored expression equal to zero and use the zero product property (if a product equals zero, at least one factor must equal zero). Setting (x - r) = 0 gives x = r, and setting (x - s) = 0 gives x = s, so the zeros are r and s—these are the x-intercepts of the parabola at points (r, 0) and (s, 0) where the graph crosses the x-axis. For P(x) = x² - 8x + 15, find numbers multiplying to 15 adding to -8: -3 and -5, so (x - 3)(x - 5); break-even at x = 3 and x = 5—positive zeros make sense for profits! Choice A correctly factors and identifies the break-even points. Choice B flips signs, giving positive factors that expand to x² + 8x + 15—signs are key, opposite for negative sum! Apply this to real-world models: factors reveal intercepts directly; practice sign relationships to avoid common errors—you're building strong skills!

This question tests your ability to factor quadratic expressions and use the factored form to identify zeros (x-intercepts) of the function—essential for graphing and solving quadratic equations. Factoring a quadratic into form f(x) = a(x - r)(x - s) reveals the zeros immediately: set the factored expression equal to zero and use the zero product property (if a product equals zero, at least one factor must equal zero). Setting (x - r) = 0 gives x = r, and setting (x - s) = 0 gives x = s, so the zeros are r and s—these are the x-intercepts of the parabola at points (r, 0) and (s, 0) where the graph crosses the x-axis. For f(x) = x² + x - 6, numbers multiply to -6, add to 1: 3 and -2, so (x - 2)(x + 3); zeros x = 2 and x = -3, x-intercepts (2, 0) and (-3, 0)—verifies the given zeros! Choice A correctly factors and states the x-intercepts. Choice B swaps the signs, expanding to x² - x - 6—middle term sign flips; verify by plugging in zeros. Test proposed zeros in original: f(2) = 4 + 2 - 6 = 0, f(-3) = 9 - 3 - 6 = 0—perfect check; build confidence with this method!

This question tests your ability to factor quadratic expressions and use the factored form to identify zeros (x-intercepts) of the function—essential for graphing and solving quadratic equations. Factoring a quadratic into form f(x) = a(x - r)(x - s) reveals the zeros immediately: set the factored expression equal to zero and use the zero product property (if a product equals zero, at least one factor must equal zero). Setting (x - r) = 0 gives x = r, and setting (x - s) = 0 gives x = s, so the zeros are r and s—these are the x-intercepts of the parabola at points (r, 0) and (s, 0) where the graph crosses the x-axis. For s(x) = x² - 7x + 10, numbers multiply to 10, add to -7: -2 and -5, so (x - 2)(x - 5); zeros x = 2 and x = 5, x-intercepts (2, 0) and (5, 0)—both positive! Choice A correctly factors and determines the x-intercepts. Choice B uses positive factors, expanding to x² + 7x + 10—signs opposite for negative sum; expand to confirm. For positive c and negative b, both factors negative; this pattern helps predict roots—great job practicing!

This question explores factoring difference of squares with coefficients, finding zeros and intercepts—perfect for graphing scaled parabolas. 4x² - 9 = (2x)² - 3² = (2x - 3)(2x + 3). Zeros from each: 2x-3=0 → x=3/2; 2x+3=0 → x=-3/2. Intercepts at (3/2,0), (-3/2,0). Symmetric and scaled! Choice B correctly factors and lists zeros/intercepts. Choice D ignores the 4: (x-3)(x+3)=x²-9, expands wrong; include the coefficient! General: a²x² - b² = (a x - b)(a x + b). Example: 9x² - 16 = (3x-4)(3x+4), zeros ±4/3. Check by expanding. Great job!

This question tests your ability to recognize and factor a difference of squares, then use the factored form to find zeros and x-intercepts for sketching the parabola. Factoring a quadratic like x² - c (difference of squares) into (x - √c)(x + √c) reveals the zeros immediately via the zero product property. Setting each factor to zero gives the roots, which are symmetric around zero for this form. These zeros are the x-intercepts, key points for graphing the parabola that opens upward from the origin. Great job spotting this special case—it simplifies everything! To factor x² - 25: recognize it as (x)² - (5)² = (x - 5)(x + 5). Set to zero: x - 5 = 0 so x = 5, or x + 5 = 0 so x = -5. Zeros are x = 5 and x = -5, with x-intercepts (5, 0) and (-5, 0). This symmetry makes sense since the vertex is at x=0. Choice B correctly identifies the zeros and intercepts from the proper factoring. Choice A mistakenly uses ±25, perhaps confusing with x² - 625 = (x - 25)(x + 25), but always check by expanding back: (x - 5)(x + 5) = x² - 25, perfect match! For difference of squares a² - b² = (a - b)(a + b), zeros at x = b and x = -b. Example: x² - 16 = (x - 4)(x + 4), zeros ±4. If it's x² + c, no real factors, but here it's minus. Remember to factor out common factors first if present, like 4x² - 9 = (2x - 3)(2x + 3). Keep practicing these patterns—they speed up solving!

This question assesses factoring perfect square trinomials and recognizing double roots, which affect the graph's behavior at the x-intercept. A perfect square trinomial factors to (x - r)², showing a double zero at x = r, meaning the parabola touches the x-axis at one point without crossing—great for understanding multiplicity! The zero product property still applies, but both factors give the same root. This x-intercept is where the vertex touches the axis if it's a minimum or maximum. You're building strong skills here! To factor x² - 6x + 9: notice it's (x - 3)² since middle term is -2*3 and constant is 3². Set (x - 3)² = 0, double root at x = 3. X-intercept: (3, 0). The factored form shows it touches and bounces. Choice B correctly factors as a square and notes the double root. Choice C has a sign error: (x + 3)² expands to x² + 6x + 9, not matching the -6x; always verify by expanding! To spot perfect squares: check if middle coefficient is twice the square root of constant with matching sign. Example: x² + 10x + 25 = (x + 5)², zero x = -5 (double). For non-squares, use general factoring. Remember multiplicity: double root means tangent to x-axis. Keep going—you've got this!

This question applies factoring to a real-world model, finding break-even points as zeros of the profit function—super relevant for business math! Factoring reveals where P(x)=0, meaning no profit or loss. Zeros indicate the production levels to break even. This connects algebra to practical decisions. Awesome work exploring applications! To factor x² - 8x + 15: find numbers multiplying to 15, adding to -8: -3 and -5. (x - 3)(x - 5). Zeros: x=3, x=5. Break-even at these units sold. The parabola opens up, profit between roots. Choice A correctly factors and identifies break-evens. Choice B has sign errors: (x+3)(x+5)=x²+8x+15, wrong signs; check expansion! For profit models ax² + bx + c, zeros show break-evens if a>0, loss outside roots. Example: x² - 10x + 24 = (x-4)(x-6), break-even 4 and 6. Interpret in context: sell between for profit. You're making great connections!

This question verifies given zeros by factoring, confirming they satisfy f(x)=0—a great check for understanding roots. Factoring should include the given roots as (x - 2)(x - (-3)) = (x-2)(x+3). This matches the quadratic. Zeros are where factors are zero. Nice verification practice! For x² + x - 6: numbers multiply -6, add 1: 3 and -2. (x + 3)(x - 2). Plug in x=2: (2)+3=5, (2)-2=0, product 0. x=-3: -3+3=0, -3-2=-5, product 0. Yes! Choice B correctly shows the factored form revealing the zeros. Choice A swaps signs: (x+2)(x-3)=x² -x -6, wrong middle term; test by expanding! To write from zeros r,s: f(x)=(x-r)(x-s). Example: zeros 1,4: (x-1)(x-4)=x²-5x+4. Verify by plugging back. You're mastering this!

This question tests solving equations by factoring and relating to function zeros (x-intercepts)—builds equation-solving skills. Factor 3x² - 12 = 0 → 3(x² - 4) = 0 → 3(x - 2)(x + 2) = 0. Zeros: ±2 (3≠0). X-intercepts: (-2, 0), (2, 0). You're doing great! Choice A correctly identifies zeros. Choice B mistakes the constant; -12/3 = -4, but √4=2, not 4. Factor completely! Difference of squares after factoring out common factor. Example: 2x² - 8 = 0 → 2(x² - 4)=0 → 2(x-2)(x+2)=0, zeros ±2.