Deriving the Quadratic Formula

5 min

Narrative

This Warm-up reminds students of the structure that governs the relationship between perfect squares written as squared factors and their equivalent expression in standard form. The key goal is for students to see that when we expand a squared factor of the form (kx+m)2(kx+m)^2, the equivalent expression in standard form has this structure: (kx)2+2kmx+m2(kx)^2 + 2kmx+ m^2. Seeing the coefficient of the squared term as something2\text {something}^2 and the coefficient of the linear term as 2 times a product of two numbers (kk and mm) enables students to complete the square more easily. This, in turn, helps them make sense of where the quadratic formula comes from, which they will explore in this lesson.

Students practice looking for and making use of structure as they think about the relationships between equivalent expressions (MP7).

Launch

Ask students to expand the squared factor (x+4)2(x+4)^2 into standard form and show all the steps. Students are likely to show (x+4)(x+4)=x2+4x+4x+42=x2+8x+16(x+4)(x+4) = x^2 + 4x + 4x + 4^2 = x^2 + 8x + 16, or (x+4)(x+4)=x2+2(4x)+42=x2+8x+16(x+4)(x+4) = x^2 + 2(4x) + 4^2 = x^2 + 8x + 16.

Next, arrange students in groups of 2. Tell students to use the relationship between the factored and standard forms to write some equivalent perfect-square expressions. Ask students to think quietly about the expressions in the table before conferring with their partner.

Student Task

Here are some perfect squares in factored and standard forms, and expressions showing how the two forms are related.

  1. Complete the table.

    factored form standard form
    (3x−4)2(3x-4)^2 (3x)2+2(‾x)(‾)+(‾)2(3x)^2 + 2(\underline{\hspace{0.2in}}x)(\underline{\hspace{0.2in}}) + (\underline{\hspace{0.2in}})^2 9x2−24x+169x^2 - 24x + 16
    (5x+‾)2(5x+\underline{\hspace{0.3in}})^2 (‾x)2+2(‾x)(‾)+(‾)2(\underline{\hspace{.2in}}x)^2 + 2(\underline{\hspace{0.2in}}x)(\underline{\hspace{0.2in}}) + (\underline{\hspace{0.2in}})^2 25x2+30x+‾25x^2 +30x + \underline{\hspace{0.3in}}
    (kx+m)2(kx+m)^2 (‾x)2+2(‾x)(‾)+(‾)2(\underline{\hspace{.2in}}x)^2 + 2(\underline{\hspace{0.2in}}x)(\underline{\hspace{0.2in}}) + (\underline{\hspace{0.2in}})^2 ‾x2+‾x+‾\underline{\hspace{0.2in}}x^2 + \underline{\hspace{0.2in}}x + \underline{\hspace{0.2in}}  
  2. Look at the expression in the last row of the table. If ax2+bx+cax^2 + bx+c is equivalent to (kx+m)2(kx+m)^2, how are a,ba, b, and cc related to kk and mm?

Sample Response

  1.  
    factored form standard form
    (3x−4)2(3x-4)^2 (3x)2+2(3x)(-4)+(-4)2(3x)^2 + 2(3x)(\text-4) + (\text-4)^2 9x2−24x+169x^2 - 24x + 16
    (5x+3)2(5x+3)^2 (5x)2+2(5x)(3)+32(5x)^2 + 2(5x)(3) + 3^2 25x2+30x+925x^2 +30x + 9
    (kx+m)2(kx+m)^2 (kx)2+2(kx)(m)+m2(kx)^2 + 2(kx)(m) + m^2 k2x2+2kmx+m2k^2x^2 + 2kmx + m^2  
  2. Sample response: a=k2a = k^2, b=2kmb = 2km, and c=m2c=m^2.

Synthesis

Display the completed table for all to see. Invite students to share their solutions for the last question. Then, ask how these values can be used to complete the square.

  • “When making 25x2+30x25x^2 + 30x a perfect square, how do you know what values to write in the middle column?” (The first squared term is (something)2(\text{something)}^2 and equals 25x225x^2, so that “something” must be 5x5x because (5x)2=25x2(5x)^2 = 25x^2. The linear term is 2(5x)(another thing)2(5x)(\text {another thing}) and equals 30x30x, so that other thing must be 3 because 2(5x)(3)=30x2(5x)(3)=30x. The last squared term is (another thing)2\text{(another thing)}^2, so it is 323^2.)

Emphasize the general structure of the equivalent expressions, as shown in the last row. When (kx+m)2(kx+m)^2 is expanded, the standard form always has the structure of (kx)2+2(kx)m+m2(kx)^2 + 2(kx)m + m^2. Also, make sure students understand how the numbers in the factored form are related to ax2+bx+cax^2+bx+c—that is, that they recognize that a=k2a = k^2, b=2kmb = 2km, and c=m2c=m^2.

Highlight that when we want to complete the square for ax2+bxax^2+bx and write an equivalent expression of the form (kx+m)2(kx+m)^2, having a perfect square for aa makes it much easier to find kk, and having an even number for bb makes it easier to find mm (because an even number gives a whole number when divided by 2).

Standards
Addressing
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • HSA-SSE.A.2·Use the structure of an expression to identify ways to rewrite it. <span>For example, see <span class="math">\(x^4 - y^4\)</span> as <span class="math">\((x^2)^2 - (y^2)^2\)</span>, thus recognizing it as a difference of squares that can be factored as <span class="math">\((x^2 - y^2)(x^2 + y^2)\)</span>.</span>
Building Toward
  • A-REI.4.a·Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.
  • A-REI.4.a·Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.
  • A-REI.4.a·Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.
  • HSA-REI.B.4.a·Use the method of completing the square to transform any quadratic equation in <span class="math">\(x\)</span> into an equation of the form <span class="math">\((x - p)^2 = q\)</span> that has the same solutions. Derive the quadratic formula from this form.

15 min

15 min