Quadratic equations with two solutions
Use this page when you want to understand why some quadratics give two different real answers and how to spot that case early.
Start here if you want the short version before reading the full method.
- A quadratic has two distinct real solutions when the discriminant b^2 - 4ac is positive.
- On a graph, that means the parabola crosses the x-axis at two different points.
What this topic means and what to look for first.
This is the most common classroom case for quadratics that factor cleanly or produce two separate roots through the quadratic formula.
Understanding the discriminant gives you a quick way to predict the outcome before finishing the whole solve.
One reliable route through the topic.
- 1Write the quadratic in standard form and identify a, b, and c.
- 2Compute the discriminant b^2 - 4ac.
- 3If it is positive, continue solving knowing there should be two different real roots.
- 4Check that the final answers are distinct and both satisfy the original equation.
See the method in action.
x^2 + 5x + 6 = 0
- The discriminant is 25 - 24 = 1, which is positive.
- That means the equation has two distinct real roots.
- Factorising confirms the roots are x = -2 and x = -3.
x^2 - 4x + 3 = 0
- The discriminant is 16 - 12 = 4, which is positive.
- So there should be two distinct real roots.
- The factorisation (x - 1)(x - 3) = 0 gives x = 1 and x = 3.
Things that commonly send the method off track.
- Finding two roots but never checking whether they are actually different values.
- Assuming a positive discriminant guarantees easy factorisation over integers.
- Mixing up the graph meaning and saying the curve only touches the x-axis once.
Use a short verification pass before moving on.
- Make sure the two roots are distinct values rather than the same repeated root.
- Check both roots by substitution or expansion.
- If you graph the quadratic, confirm that it crosses the x-axis twice.
Try a few variations before switching to a calculator or solver tool.
- x^2 - 7x + 10 = 0
- 2x^2 - x - 6 = 0
- x^2 + x - 12 = 0
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Extra algebra revision resources
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Short answers worth checking.
A positive discriminant means there are two distinct real roots.
No. Some quadratics have two real roots but do not factor neatly over simple integers.
Continue with the next closely related topic.
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