What the Discriminant Tells You

b2 - 4ac sorts every quadratic into one of three cases before any solving starts: two real roots, one repeated root or none.

For ax2 + bx + c = 0 the discriminant is b2 - 4ac. It is computed first, and its sign alone fixes how many real roots exist, which saves minutes spent hunting for factor pairs that do not exist.

Positive case: x2 + 5x - 14 = 0 gives 25 + 56 = 81. Positive, and a perfect square, so there are two real roots and they come out whole: x = 2 and x = -7.

Zero case: x2 - 6x + 9 gives 36 - 36 = 0. A zero discriminant means one repeated root: the equation is (x - 3)2 = 0 and the only solution is x = 3.

Negative case: x2 + x + 1 gives 1 - 4 = -3. Negative means no real roots: the parabola sits entirely above the x-axis, and 'no real solutions' is the complete answer.

A quadratic has exactly two roots in total, so the only possibilities are two, one or zero real solutions, never three. Once the case is known, the full solving methods are on the quadratic page, and the same checking habit carries over to the simultaneous page.

• positive: two distinct real roots
• zero: one repeated real root
• negative: no real roots
• perfect square: the roots come out rational

Further reading

• How to Solve a Quadratic Equation
• Solving Simultaneous Equations by Elimination