Math 2–3 Honors · Lesson Review · October 8, 2026

The Quadratic Formula and Discriminant

Derive the quadratic formula by completing the square, solve quadratics with exact roots, and use the discriminant to predict two, one, or no real solutions.

28-minute edited lesson8 chaptersCaptions + transcript

01 · Watch

Lesson video

VTT captions SRT captions

0:0001Deriving the quadratic formula 6:3502Standard form and two integer roots 10:3503A mnemonic for the formula 11:3204Exact roots and factorization 15:1905Simplify a common factor 18:3606One repeated real root 21:3207When there are no real roots 23:2508The discriminant classifies roots

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02 · Understand

Learning targets

Derive the quadratic formula from completing the square.
Write a quadratic in standard form and substitute signed coefficients correctly.
Simplify exact rational and irrational roots without dropping the ±.
Use the discriminant to predict the number of distinct real solutions.

03 · Connect

Key ideas

The recording derives the formula, applies it to five classroom examples, and then connects the discriminant to real solutions.

01Completing the square produces the formula: $ax^2+bx+c=0\quad\Longrightarrow\quad(2ax+b)^2=b^2-4ac$ For a≠0, multiply the equation by 4a, move 4ac, and add b². Taking both square roots and then dividing by 2a gives the quadratic formula. This identity also works when a is negative.
02Standard form comes first: $x^2-4=3x\quad\Longrightarrow\quad x^2-3x-4=0$ Move every term to one side before identifying a, b, and c. A coefficient includes its sign. In this example a=1, b=−3, and c=−4.
03The entire numerator is divided by 2a: $x=\frac{-b+\sqrt D}{2a}\quad\text{or}\quad x=\frac{-b-\sqrt D}{2a}$ The ± represents two cases when D>0. Keep the radical and −b together in the numerator. Simplify common factors only when they divide every numerator term and the denominator.
04The discriminant counts distinct real roots: $D=b^2-4ac:\quad D>0\Rightarrow2,\quad D=0\Rightarrow1,\quad D<0\Rightarrow0$ These counts refer to distinct real roots of a real-coefficient quadratic with a≠0. When D=0 the same root occurs twice. When D<0 there are no real roots; complex solutions are a separate topic.
05Exact roots reveal the factors: $ax^2+bx+c=a(x-r_1)(x-r_2)$ For two real roots, include the leading coefficient when reconstructing the polynomial. With integer coefficients, a positive perfect-square discriminant gives rational roots; a positive nonsquare discriminant gives irrational roots. Keep exact radicals rather than rounded decimals.

04 · Apply

Worked examples

These examples follow the methods developed in the recorded lesson.

Classroom example 1

Move the linear term before substituting

x^2-4=3x\quad\Longrightarrow\quad x^2-3x-4=0

Read a=1, b=−3, c=−4.

D=(-3)^2-4(1)(-4)=25,\qquad x=\frac{3\pm5}{2}

Parentheses protect the negative coefficients.

x=4\text{ or }x=-1

Both values also solve the original equation.

Classroom example 2

Keep irrational solutions exact

2x^2-5x-1=0,\qquad a=2,\ b=-5,\ c=-1

The leading coefficient is not one.

D=(-5)^2-4(2)(-1)=33,\qquad x=\frac{5\pm\sqrt{33}}4

The radical cannot be simplified further.

2x^2-5x-1=2\left(x-\frac{5+\sqrt{33}}4\right)\left(x-\frac{5-\sqrt{33}}4\right)

The outside factor 2 preserves the original leading coefficient.

Classroom example 3

Simplify every term in a radical fraction

2x^2+4x-2=0,\qquad D=4^2-4(2)(-2)=32

Use a=2, b=4, c=−2.

x=\frac{-4\pm\sqrt{32}}4=\frac{-4\pm4\sqrt2}4

Extract the perfect-square factor 16.

x=-1\pm\sqrt2

Both numerator terms and the denominator share a factor of 4.

Classroom example 4

Recognize a repeated root

x^2-10x+25=0,\qquad D=(-10)^2-4(1)(25)=0

A zero discriminant makes the two formula cases identical.

x=\frac{10\pm0}{2}=5,\qquad x^2-10x+25=(x-5)^2

There is one distinct real root, with multiplicity two.

Classroom example 5

Identify no real solutions before simplifying

x^2+2x+10=0,\qquad D=2^2-4(1)(10)=-36

The value under the radical is negative.

\text{Real solution set: }\varnothing

No real number squares to −36. State no real solutions precisely.

05 · Avoid

Common mistakes

Reading coefficients before setting the equation equal to zero.

Move all terms to one side first; otherwise the signs of b or c can be wrong.

Squaring a negative coefficient without parentheses.

For b=−5, b²=(−5)²=25, and −b=5. These are different operations.

Dividing only the radical by 2a.

The fraction bar covers the whole numerator, −b±√D.

Cancelling part of a sum.

A cancelled factor must multiply every numerator term as well as the denominator.

Calling every quadratic's solutions two real roots.

D=0 gives one repeated real root; D<0 gives no real roots.

06 · Check yourself

Try it first

Use these five fresh problems to check your understanding. Open a solution after you try it.

1Put in standard form and solve using the quadratic formula:

3x^2=5x+2

$3x^2-5x-2=0,\qquad D=(-5)^2-4(3)(-2)=49$

Use a=3, b=−5, c=−2.

$x=\frac{5\pm7}{6}=2\text{ or }-\frac13$

Evaluate both signs and reduce the fractions.

2Find both exact roots:

x^2+6x+2=0

$D=6^2-4(1)(2)=28,\qquad x=\frac{-6\pm\sqrt{28}}2$

The positive nonsquare discriminant gives two irrational real roots.

$x=\frac{-6\pm2\sqrt7}2=-3\pm\sqrt7$

Simplify the radical, then divide both terms by 2.

3Use the discriminant to count the distinct real roots, then solve:

4x^2-12x+9=0

$D=(-12)^2-4(4)(9)=0$

There is one distinct real root.

$x=\frac{12}{8}=\frac32,\qquad4x^2-12x+9=(2x-3)^2$

The same root is repeated.

4Determine the real solution set without using decimals:

2x^2-4x+5=0

$D=(-4)^2-4(2)(5)=-24$

A negative discriminant rules out real roots.

$\text{Real solution set: }\varnothing$

The quadratic formula would require a square root of a negative number.

5For real k, classify the number of distinct real roots. Then solve when k=5:

x^2-6x+k=0

$D=36-4k$

Compare this expression with zero.

$k<9:\ 2\text{ real roots};\quad k=9:\ 1;\quad k>9:\ 0$

The repeated root when k=9 is x=3.

$k=5:\quad x=\frac{6\pm\sqrt{16}}2=5\text{ or }1$

The five practice prompts differ from the recording's classroom examples.

07 · Revisit

Transcript and captions

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