Algebra 1 PARCC question: quadratic zeros

Correct answers: –7 and 2.

PARCC evidence statement(s) tested: A.REI.4b-1:

Solve quadratic equations in one variable. … Solve quadratic equations with rational number coefficients by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.

i) Tasks should exhibit variety in initial forms. Examples of quadratic equations with real solutions: t² = 49, 3a² = 4, 7 = x², r² = 0, ½y² = 1/5, y² – 8y + 15 = 0, 2x² – 16x + 30 = 0, 2p = p² + 1, t² = 4t, 7x² + 5x – 3 = 0, ¾c(c – 1) = c, (3c − 2)² = 6x − 4

ii) Methods are not explicitly assessed; strategy is assessed indirectly by presenting students with a variety of initial forms.

iii) For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required; however, students will not be penalized if they simplify the radicals correctly.

iv) Prompts integrate mathematical practices by not indicating that the equation is quadratic. (e.g., “Find all real solutions of the equation t² = 4t” … not, “Solve the quadratic equation t² = 4t”).

The evidence statement above references Math Practice 5 and Math Practice 7 in the Common Core: [5] Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. … For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. …

[7] Mathematically proficient students look closely to discern a pattern or structure. … Students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. … They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

The question tests students’ understanding of the high school Common Core algebra standard HSA.SSE.B.3, found under high school algebra (seeing structure in expressions), which states that they should be able to “choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression [and, 3.A] factor a quadratic expression to reveal the zeros of the function it defines.”

Solve a quadratic equation for x (hopefully by factoring).

I first notice the overall structure of the equation: I have x², x, and a constant on each side, and so like terms can be combined easily. Squaring the binomial on the left, we get

Combining like terms and moving everything over to the left side, we quickly get

I can (thankfully) factor the left side by noting that –2 × 7 = –14 and –2 + 7 = 5.

Once we get to this point, we know from the multiplicative property of 0 that if a × b = 0, then either a = 0, b = 0, or both a and b = 0.

So if (x–2) = 0, then x obviously is 2, and if (x+7) = 0, then x must be –7.

All solutions should be checked, which I can do in any number of ways. If I have a graphing calculator (or Mathematica) handy, I can plot the quadratic function on a coordinate grid and see where the graph intersects the x axis. It should be at the solutions I found for x.

If I don’t have the means to graph the function, say during the PARCC test, I can plug the values I found for x back into the equation and make sure it’s true.