Lesson 18: Applying the Quadratic Formula (Part 2)

Applying the Quadratic Formula (Part 2)

5 min

Narrative

In this Warm-up, students evaluate variable expressions that resemble those in the quadratic formula. The aim is to preview the calculations that are the source of some common errors in solving equations using the quadratic formula, which students will analyze in the next activity.

Student Task

Evaluate each expression for $a=9$ , $b=\text-5$ , and $c=\text-2$

$\text-b$
$b^2$
$b^2-4ac$
$\text-b \pm \sqrt{a}$

Sample Response

5
25
97
8 and 2

Activity Synthesis (Teacher Notes)

Invite students to share their responses and discuss any disagreement. For each expression, ask if they can think of an error someone might make when evaluating such an expression. Some possible errors:

$\text-b$ : Forgetting that $\text-b$ is really $\text-1(b)$ and the product of two negative numbers is positive.
$b^2$ : When $b$ is -5, evaluating $\text-(5)^2$ , instead of $(\text-5)^2$ .
$b^2-4ac$ : Forgetting that subtracting by $4ac$ is equivalent to adding $\text-4ac$ , or neglecting to see that if $c$ is negative, $\text-4ac$ is positive, not negative.
$\text-b \pm \sqrt{a}$ : Neglecting the negative in front of $b$ , or neglecting to see that the expression takes two different values. Tell students that in the next activity, they will spot some errors in solving quadratic equations.

Standards

Building On

6.EE.2.c·Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.
6.EE.A.2.c·Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas $V = s^3$ and $A = 6 s^2$ to find the volume and surface area of a cube with sides of length $s = 1/2$.
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 $x^2 = p$ and $x^3 = p$, where $p$ is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that $\sqrt{2}$ is irrational.

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 $x^2 = 49$), 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 $a \pm bi$ for real numbers $a$ and $b$.

15 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Applying the Quadratic Formula (Part 2)

5 min

Narrative

Student Task

Evaluate each expression for $a=9$ , $b=\text-5$ , and $c=\text-2$

$\text-b$
$b^2$
$b^2-4ac$
$\text-b \pm \sqrt{a}$

Sample Response

5
25
97
8 and 2

Activity Synthesis (Teacher Notes)

$\text-b$ : Forgetting that $\text-b$ is really $\text-1(b)$ and the product of two negative numbers is positive.
$b^2$ : When $b$ is -5, evaluating $\text-(5)^2$ , instead of $(\text-5)^2$ .
$b^2-4ac$ : Forgetting that subtracting by $4ac$ is equivalent to adding $\text-4ac$ , or neglecting to see that if $c$ is negative, $\text-4ac$ is positive, not negative.
$\text-b \pm \sqrt{a}$ : Neglecting the negative in front of $b$ , or neglecting to see that the expression takes two different values. Tell students that in the next activity, they will spot some errors in solving quadratic equations.

Standards

Building On

6.EE.2.c·Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.
6.EE.A.2.c·Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas $V = s^3$ and $A = 6 s^2$ to find the volume and surface area of a cube with sides of length $s = 1/2$.
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 $x^2 = p$ and $x^3 = p$, where $p$ is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that $\sqrt{2}$ is irrational.

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 $x^2 = 49$), 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 $a \pm bi$ for real numbers $a$ and $b$.

Applying the Quadratic Formula (Part 2)

Warm-upBits and Pieces5 minKCs

Narrative

Student Task

Sample Response

ActivityUsing the Formula with Care15 minKCs

ActivitySure about That?15 minKCs

Knowledge Components

Applying the Quadratic Formula (Part 2)

Warm-upBits and Pieces5 minKCs

Narrative

Student Task

Sample Response

ActivityUsing the Formula with Care15 minKCs

ActivitySure about That?15 minKCs

5 min

15 min

15 min

5 min

15 min

15 min