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 formula to find the discriminant in a quadratic equation?
For a quadratic written as ax² + bx + c = 0, the discriminant is D = b² − 4ac. It is the expression under the square root in the quadratic formula. Before you plug in, make sure the equation is in standard form with everything on one side, and keep the signs of a, b and c exactly as they appear.
How do you calculate the discriminant of a quadratic equation?
Identify a, b and c, then compute b² − 4ac. For 2x² + 5x + 3 = 0, a = 2, b = 5 and c = 3, so D = 25 − 24 = 1. Since 1 is positive, the equation has two real solutions. Square b first, then subtract 4ac, and be careful when b is negative: (−5)² is 25, not −25.
How does the discriminant show the nature of the roots?
The sign of D tells you what kind of roots to expect. If D > 0, there are two different real roots; if D is also a perfect square, they are rational. If D = 0, there is one repeated real root and the parabola touches the x-axis at its vertex. If D < 0, there are no real roots, only two complex ones.
What is the discriminant of 2x² + 4x + 3 = 0?
Here a = 2, b = 4 and c = 3, so D = 4² − 4(2)(3) = 16 − 24 = −8. A negative discriminant means the equation has no real solutions, so the graph of y = 2x² + 4x + 3 stays entirely above the x-axis. You would only get complex roots if you continued with the quadratic formula.
What is the discriminant of 5x² + 5x − 6 = 0?
With a = 5, b = 5 and c = −6, D = 5² − 4(5)(−6) = 25 + 120 = 145. Subtracting a negative number adds, which is the step students most often miss. Because 145 is positive but not a perfect square, the equation has two real roots that are irrational, so it will not factor nicely over the integers.
Is there a worksheet for finding the discriminant of a quadratic equation?
Yes. A good discriminant worksheet mixes equations with positive, zero and negative values of D so you practice reading all three cases. Try including ones like x² − 6x + 9 = 0, where D = 36 − 36 = 0, and ones with a negative c. Use the worksheet maker on the page to build a fresh set with answers.
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