A.8(B)·Algebra II·Quadratic & Square Root FunctionsSupporting

Quadratic Formula and Discriminant

Solve quadratic equations with the quadratic formula and interpret the discriminant.

Students rewrite quadratic equations in standard form, interpret the discriminant, and use the quadratic formula to obtain exact solutions.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice50 minutes

Learn the concept

Standard form first

The quadratic formula applies to $ax^2+bx+c=0$ , with $a\ne 0$ . Move every term to one side before identifying the signed values of a, b, and c.

Interpret the discriminant

The discriminant is $D=b^2-4ac$ . If $D>0$ , there are two distinct real roots; if $D=0$ , there is one repeated real root; if $D<0$ , there are two nonreal complex roots.

Simplify both solutions

Use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ . The entire numerator is divided by $2a$ . Simplify square factors inside the radical, then reduce common factors.

Worked example

Find the exact solution set of $x^2+7=8x$ .

Rewrite as $x^2-8x+7=0$ , giving $a=1$ , $b=-8$ , and $c=7$ .
Compute $D=(-8)^2-4(1)(7)=36>0$ , so there are two real roots.
Substitute: $x=\frac{8\pm\sqrt{36}}{2}=\frac{8\pm6}{2}$ .
Evaluate both branches: $x=7$ or $x=1$ .

$\{1,7\}$

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Solving Quadratic Equations

UT IPSI · Kid2Kid / Texas Education Agency

Solve quadratic equations and compare exact solution methods.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. Solve $x^2+2=4x$ exactly.
Show answer and reasoning
$x=2\pm\sqrt{2}$
Standard form is $x^2-4x+2=0$ . Thus $D=8$ and $x=\frac{4\pm2\sqrt2}{2}$ .
2. Classify the solutions of $2x^2+4x+5=0$ .
Show answer and reasoning
Two nonreal complex solutions; no real solutions.
$D=4^2-4(2)(5)=-24<0$ .

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 35 · 1 point

What is the solution set for $x^2 + 4 = 6x$ ?

$\left\{-3 \pm i\sqrt{14}\right\}$
$\left\{-3 \pm \sqrt{5}\right\}$
$\left\{3 \pm i\sqrt{14}\right\}$
$\left\{3 \pm \sqrt{5}\right\}$

Show STAAR answer and explanation

$\left\{3 \pm \sqrt{5}\right\}$

Rewrite as $x^2-6x+4=0$ . The discriminant is $(-6)^2-4(1)(4)=20>0$ , so there are two real solutions: $x=\frac{6\pm\sqrt{20}}{2}=3\pm\sqrt5$ .

$\left\{-3 \pm i\sqrt{14}\right\}$ : This uses the wrong sign for $-b$ and treats the discriminant as negative; $b^2-4ac=20$ is positive.

$\left\{-3 \pm \sqrt{5}\right\}$ : With $b=-6$ , $-b=6$ , so the solutions are centered at 3, not −3.

$\left\{3 \pm i\sqrt{14}\right\}$ : The discriminant is +20, not negative, so the solutions are real, not imaginary.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Identify signed coefficients after setting an equation equal to zero.
Use the discriminant to classify solutions.
Simplify quadratic-formula results with irrational roots.

Instructional sequence

1
Engage
5 min
Compare equations that factor over integers with ones that do not.
2
Direct instruction
12 min
Model standard form, signed substitution, and discriminant interpretation.
3
Guided practice
13 min
Solve equations with positive nonsquare discriminants and simplify radicals.
4
Error analysis
13 min
Correct missing negative signs and denominators that divide only one term.
5
Closure
7 min
Students classify and solve the exit-ticket equation.

Misconceptions to address

““b is positive whenever the original equation has a positive x-term.””

Teacher move: Move all terms to one side first. Then read each coefficient with its sign.

““A positive nonsquare discriminant gives imaginary roots.””

Teacher move: A positive discriminant gives real roots; a nonsquare positive discriminant gives irrational real roots.

Materials

•Quadratic-formula reference
•Perfect-square chart
•Scientific calculator

Teaching tips

Write a, b, and c on a separate line.
Use parentheses around negative coefficients in every substitution.
Keep exact radical answers before checking decimal approximations.

Exit ticket · 3–5 min

Solve $x^2-4x+1=0$ exactly.

Show expected answer

$D=16-4=12$ , so $x=\frac{4\pm\sqrt{12}}{2}=2\pm\sqrt{3}$ .

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Standard form first

Interpret the discriminant

Simplify both solutions

Worked example

Watch the lesson

Algebra 2 Solving Quadratic Equations

Try it yourself

1. Solve x2+2=4xx^2+2=4xx2+2=4x exactly.

2. Classify the solutions of 2x2+4x+5=02x^2+4x+5=02x2+4x+5=0.

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 35 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Error analysis

Closure

Misconceptions to address

1. Solve $x^2+2=4x$ exactly.

2. Classify the solutions of $2x^2+4x+5=0$ .