How to Find Discriminant of Quadratic Equation?
Finding Discriminant of Quadratic Equation
The discriminant, \(b^2 – 4ac\), is the part of the quadratic formula under the square root — and its sign tells you how many real solutions a quadratic has before you solve. Positive means two, zero means one, negative means none. We’ll read it fast, with a solver and a worksheet maker a tap away.
Find Discriminant of Quadratic Equation: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Read the formFactored, standard, and vertex forms reveal different features.
- Choose the methodFactor when friendly, complete the square for structure, or use the formula when needed.
- Connect to the graphRoots are x-intercepts and the vertex is the minimum or maximum point.
Worked examples
Factor and solve
- Factor into (x – 3)(x – 4).
- Set each factor equal to zero.
- Solve both small equations.
Find the axis
- Use x = -b/(2a).
- Here a = 2 and b = -8.
- Compute 8/4.
Try one before moving on
Find Discriminant of Quadratic Equation: pop-up practice
The discriminant is a small but powerful piece of the quadratic formula: the expression \(b^2 – 4ac\) tucked under the square root. Its real value is that its sign tells you how many real solutions a quadratic has — before you do any solving. One quick calculation answers “two, one, or none?”
In short: the discriminant is \(D = b^2 – 4ac\). If \(D > 0\) there are two real solutions; if \(D = 0\), exactly one; if \(D < 0\), no real solutions.
The Sign Tells the Story
In the quadratic formula \(x = \tfrac{-b \pm \sqrt{b^2 – 4ac}}{2a}\), everything hinges on the square root. A positive inside gives two different real roots; a zero gives one (the \(\pm\) adds nothing); a negative means the square root isn’t real, so there are no real solutions.
Worked Examples
The sign of \(D\) predicts the graph: two crossings, one touch, or no contact — see each below.
Example A — Two solutions
Discriminant of \(x^2 – 5x + 6\)?
- Identify \(a=1,\ b=-5,\ c=6\).
- \(D = (-5)^2 – 4(1)(6) = 25 – 24 = 1\).
- \(D > 0\): two real solutions — the parabola crosses the x-axis twice.
Answer: \(D = 1\), two real
Example B — One solution
Discriminant of \(x^2 – 4x + 4\)?
- \(a=1,\ b=-4,\ c=4\).
- \(D = 16 – 16 = 0\).
- \(D = 0\): one repeated root — the parabola just touches the axis at its vertex.
Answer: \(D = 0\), one real
Example C — No real solutions
Discriminant of \(x^2 + x + 1\)?
- \(a=1,\ b=1,\ c=1\).
- \(D = 1 – 4 = -3\).
- \(D < 0\): no real solutions — the parabola floats clear of the x-axis.
Answer: \(D = -3\), no real
Example D — Negative \(c\)
Discriminant of \(2x^2 + 3x – 2\)?
- \(a=2,\ b=3,\ c=-2\).
- Mind the sign: \(D = 9 – 4(2)(-2) = 9 + 16 = 25\).
- \(D > 0\): two real solutions (roots \(\tfrac12\) and \(-2\)).
Answer: \(D = 25\), two real
Why It’s Useful
The discriminant saves you from solving a quadratic that has no real answer, and it’s a fast checkpoint on multiple-choice tests. Geometrically it tells you whether a parabola crosses the x-axis twice, just touches it, or floats clear of it — handy when you only need the number of solutions, not the solutions themselves.
Slip-Ups That Cost Easy Points
- Sign of \(-4ac\). When \(c\) is negative, \(-4ac\) is positive — a frequent error that flips the result.
- Squaring \(b\) wrong. \(b^2\) is always positive: \((-5)^2 = 25\), not \(-25\).
- Confusing \(D = 0\) with “no solution.” Zero means exactly one real solution, not none.
- Forgetting to identify \(a, b, c\) first. Write them down from \(ax^2 + bx + c\) before plugging in.
Your Turn: Find \(D\) and the Count
Compute the discriminant and state the number of real solutions. Reveal to check.
- \(x^2 – 6x + 9\)
- \(x^2 + 2x + 5\)
- \(x^2 – 7x + 10\)
- \(3x^2 – 2x + 1\)
Show answers
- \(\color{blue}{D = 0 \Rightarrow \text{one}}\)
- \(\color{blue}{D = -16 \Rightarrow \text{none}}\)
- \(\color{blue}{D = 9 \Rightarrow \text{two}}\)
- \(\color{blue}{D = -8 \Rightarrow \text{none}}\)
Make Your Own Discriminant Worksheet
Generate fresh discriminant problems with a full answer key — print or save as a PDF.
Frequently Asked Questions
What is the discriminant?
It’s \(b^2 – 4ac\), the quantity under the square root in the quadratic formula. Its sign tells you how many real solutions a quadratic has.
What does each sign mean?
\(D > 0\): two real solutions. \(D = 0\): one (repeated) real solution. \(D < 0\): no real solutions (the roots are complex).
Why is the discriminant useful if I haven’t solved yet?
It tells you the number of real solutions instantly, so you know what to expect — or whether a real solution even exists — before doing the full work.
How does it relate to the graph?
\(D > 0\) means the parabola crosses the x-axis twice; \(D = 0\) means it touches once at the vertex; \(D < 0\) means it never reaches the x-axis.
Related Topics
Continue Your Study
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