Quadratic equations: factoring, the formula and the discriminant
Solving ax² + bx + c = 0 by factoring, by completing the square and by the formula; what Δ = b² − 4ac says about the roots; sum and product of the roots.
First, the rules
Three ways to solve, one number that decides
Bring everything to one side first.
Two numbers with product c and sum b (for a = 1). A product is zero only when a factor is zero.
Works every time; use it when factoring is not obvious.
Δ > 0 two different real roots · Δ = 0 one repeated root · Δ < 0 no real root.
See it
Watch Δ decide
Slide a, b, c. When Δ is positive the parabola crosses the axis twice, at zero it touches once, below zero it never arrives — the roots are read straight off the formula.
Step by step
Worked examples
The picture behind every quadratic equation is a parabola — the roots are where it crosses the x-axis. That is the next lesson; here we only solve.
Factoring
product and sumSolve x² − x − 12 = 0 and 2x² − 8x = 0.
Ask yourself: For the first: two numbers whose product is −12 and sum is −1. For the second: what is common to both terms?
Show the solution
- −4 and 3: product −12, sum −1. (x − 4)(x + 3) = 0 ⇒ x = 4 or x = −3.
- Common factor 2x: 2x(x − 4) = 0 ⇒ x = 0 or x = 4. Never divide both sides by x — you would lose the root 0.
Answer {−3, 4} and {0, 4}
The formula and the discriminant
Δ decidesSolve 2x² − 3x − 4 = 0, then decide how many real roots x² + 4x + 5 = 0 has without solving it.
Ask yourself: Compute Δ first — it tells you how many roots to expect before you write any square roots.
Show the solution
- a = 2, b = −3, c = −4: Δ = 9 + 32 = 41 > 0, two roots.
- x = 3 ± √414 (irrational, so factoring could not have worked).
- Second equation: Δ = 16 − 20 = −4 < 0: no real root (the parabola never reaches the x-axis).
Answer 3 ± √414; no real root
Without finding the roots
Vietax² − 5x + 3 = 0 has roots x₁ and x₂. Find x₁² + x₂² and the equation whose roots are 2x₁ and 2x₂.
Ask yourself: You never need the roots themselves — only their sum and product.
Show the solution
- x₁ + x₂ = 5, x₁x₂ = 3.
- x₁² + x₂² = (x₁ + x₂)² − 2x₁x₂ = 25 − 6 = 19.
- New roots: sum 2(x₁ + x₂) = 10, product 4x₁x₂ = 12, equation x² − 10x + 12 = 0.
Answer 19; x² − 10x + 12 = 0
Practice: quadratic equations
0 / 0Roots by factoring, the discriminant and the number of roots, sums and products, and building an equation from its roots. Your best score is kept in this browser only.
About this lesson
Quadratic equations: factoring, completing the square, the quadratic formula x = (−b ± √Δ) / 2a, the discriminant Δ = b² − 4ac and the three cases (two roots, one repeated root, no real root), Vieta’s formulas for the sum and product of the roots, and building an equation from its roots. Worked examples and practice with exact answers.
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