The Quadratic Formula

5 min

Narrative

This Warm-up prompts students to evaluate the kinds of numerical expressions they will see in the lesson. The expressions involve rational square roots, fractions, and the ±\pm notation.

As students work, notice any common errors or challenges so they can be addressed during the class discussion.

Launch

Tell students to evaluate the expressions without using a calculator.

Student Task

Each expression represents two numbers. Evaluate the expressions and find the two numbers.

  1. 1±491 \pm \sqrt{49}
  2. 8±25\displaystyle \frac{8 \pm 2}{5}
  3. ±(-5)2−4⋅4⋅1\pm \sqrt{(\text-5)^2-4 \boldcdot 4 \boldcdot 1}
  4. -18±362⋅3\displaystyle \frac{\text-18 \pm \sqrt{36}}{2 \boldcdot 3}

Sample Response

  1. 8 and -6
  2. 2 and 65\frac65
  3. 3 and -3
  4. -2 and -4

Synthesis

Select students to share their responses and reasoning. Address any common errors. As needed, remind students of the properties and order of operations and the meaning and use of the ±\pm symbol.

Anticipated Misconceptions

Students may be unfamiliar with evaluating rational expressions in which the numerator contains more than one term. To help students see the structure of the expressions, consider decomposing them into a sum of two fractions. For example, show that 8±25\frac{8\pm2}{5} can be written as 85±25\frac85\pm\frac25. This approach can also help to avoid a common error of dividing only the first term by the denominator (4+72≠2+7\frac{4+7}{2}\ne2+7). Some students may incorrectly write 362\frac{\sqrt{36}}{2} as 18\sqrt{18}. Point out that the first expression is equal to 3, while the other has to be greater than 3 since 18≈4.243\sqrt{18}\approx4.243.

Standards
Building On
  • 8.EE.2·Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
  • 8.EE.A.2·Use square root and cube root symbols to represent solutions to equations of the form <span class="math">\(x^2 = p\)</span> and <span class="math">\(x^3 = p\)</span>, where <span class="math">\(p\)</span> is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that <span class="math">\(\sqrt{2}\)</span> is irrational.
Addressing
  • A-SSE.A·Interpret the structure of expressions
  • HSA-SSE.A·Interpret the structure of expressions.
Building Toward
  • A-REI.4.b·Solve quadratic equations 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. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • A-REI.4.b·Solve quadratic equations 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. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • A-REI.4.b·Solve quadratic equations 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. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • HSA-REI.B.4.b·Solve quadratic equations by inspection (e.g., for <span class="math">\(x^2 = 49\)</span>), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as <span class="math">\(a \pm bi\)</span> for real numbers <span class="math">\(a\)</span> and <span class="math">\(b\)</span>.

10 min

20 min