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MathChapter 11: Quadratic Functions

Notes

Quadratic Formula and the Discriminant Notes

Apply the quadratic formula accurately, interpret the discriminant, connect root count to graphs, and use root relationships efficiently.

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The quadratic formula solves every equation \(ax^2+bx+c=0\) with \(a\ne0\). The discriminant inside its radical often answers a root-count question before any roots are calculated.

Quadratic formula workflow

Quadratic formula
\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

First rewrite the equation in standard form and preserve the signs of \(a\), \(b\), and \(c\).

Anatomy of the quadratic formula

Text labels identify the opposite linear coefficient, two solution branches, discriminant, and complete denominator.

\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
Opposite of the linear coefficient
\(-b\)

Keep parentheses when the coefficient is negative.

Two branches
\(\pm\)

Evaluate the plus and minus cases unless the radical is zero.

Discriminant
\(b^2-4ac\)

Its sign determines the number of real roots.

Entire denominator
\(2a\)

Both numerator terms are divided by twice the leading coefficient.

  1. Standardize

    Move every term to one side so the equation equals \(0\).

  2. Label

    Write \(a\), \(b\), and \(c\) with their signs.

  3. Discriminant

    Compute \(D=b^2-4ac\) separately to reduce sign errors.

  4. Substitute

    Place \(-b\), \(\pm\sqrt D\), and \(2a\) carefully.

  5. Simplify

    Simplify the radical and both branches exactly.

Worked example

Solve with exact radical answers

Solve \(3x^2+2x-7=0\).

  1. Coefficients

    \(a=3\), \(b=2\), and \(c=-7\).

  2. Discriminant

    \(D=2^2-4(3)(-7)=88\).

  3. Formula

    \(x=\frac{-2\pm\sqrt{88}}{6}=\frac{-2\pm2\sqrt{22}}{6}\).

  4. Reduce

    Divide numerator and denominator by \(2\).

\(x=\frac{-1\pm\sqrt{22}}{3}\).

The discriminant predicts the graph

Discriminant sign, real roots, and graph behavior
ConditionReal rootsGraph behavior
\(D>0\)Two distinct real rootsThe parabola crosses the horizontal axis twice.
\(D=0\)One repeated real rootThe parabola touches the horizontal axis once.
\(D<0\)No real rootsThe parabola does not meet the horizontal axis.

Three discriminant cases

Each exact quadratic is rendered from the same coefficients used to calculate its discriminant and roots.

Positive discriminant: two rootsUpward-opening parabola with vertex at 0, -4, symmetry axis at horizontal coordinate 0, 2 real horizontal-axis intercepts at -2 and 2, and vertical-axis intercept -4.-6-5-4-3-2-1123456-5-4-3-2-1123456xyvertex (0,-4)root (-2,0)root (2,0)f(x) = x^2-4
Positive discriminant: two roots
Zero discriminant: one repeated rootUpward-opening parabola with vertex at 1, 0, symmetry axis at horizontal coordinate 1, 1 real horizontal-axis intercept at 1, and vertical-axis intercept 1.-6-5-4-3-2-1123456-2-112345678xyvertex and repeated root (1,0)f(x) = x^2-2x+1
Zero discriminant: one repeated root
Negative discriminant: no real rootsUpward-opening parabola with vertex at -1, 4, symmetry axis at horizontal coordinate -1, no real horizontal-axis intercepts, and vertical-axis intercept 5.-6-5-4-3-2-1123456-1123456789101112xyvertex (-1,4)f(x) = x^2+2x+5
Negative discriminant: no real roots

Parameter and tangency questions

Worked example

Force exactly one real root

For what values of \(k\) does \(x^2+kx+16=0\) have exactly one real solution?

  1. Use the condition

    Exactly one real root requires \(D=0\).

  2. Set the discriminant

    \(k^2-4(1)(16)=0\), so \(k^2=64\).

  3. Keep both parameter values

    \(k=8\) or \(k=-8\).

\(k=\pm8\).

Sum and product of roots

Root relationships that avoid unnecessary solving
QuantityFormulaWhen useful
Sum of roots\(r_1+r_2=-\frac{b}{a}\)When only the total of the two roots is requested
Product of roots\(r_1r_2=\frac{c}{a}\)When only the product is requested
Discriminant\(D=b^2-4ac\)When root count or graph intersection behavior is requested

Common mistakes

  • Record a negative \(b\) with its sign before computing \(-b\) or \(b^2\).
  • The full numerator is divided by \(2a\), not only the radical.
  • Use \(b^2-4ac\), never \(b^2+4ac\).
  • \(D=0\) means one repeated real root, not zero roots.
  • A negative discriminant means no real roots; it does not mean one negative root.
Mini check

Count roots efficiently

How many real solutions does \(2x^2-4x+7=0\) have?

  1. Zero
  2. One
  3. Two
  4. Cannot be determined
Show answer and explanation

Answer: Zero

\(D=(-4)^2-4(2)(7)=16-56=-40<0\), so there are no real solutions.

Key takeaways

Key takeaways

What to remember

  • Standardize first and preserve coefficient signs.
  • The discriminant controls real-root count and horizontal-axis intersections.
  • Use \(-\frac ba\) and \(\frac ca\) when only root sum or product is needed.
  • Retain exact radicals unless an approximation is explicitly requested.
Continue learning

Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.