Two, One, or None

Part of the Factoring & Quadratics section of Coddy's Algebra journey. Lesson 54 of 63.

The discriminant of ax^2 + bx + c = 0 is the number under the square root in the quadratic formula, b^2 - 4ac. Its sign tells you how many real solutions the equation has, before you find any of them.

If the discriminant is positive, the plus and the minus in the formula give two different values, so there are two solutions. For x^2 + 4x + 3 = 0 the discriminant is 4^2 - 4 × 1 × 3 = 4. It is positive, so the equation has two solutions. If the discriminant is zero, adding zero and subtracting zero give the same value, so there is one solution. If the discriminant is negative, no real number squares to a negative number, so there are no real solutions.

The number of solutions also shows in the graph of y = ax^2 + bx + c, which is a curve called a parabola. When there are two solutions, the parabola crosses the x-axis at two points. When there is one solution, it touches the x-axis at one point. When there are no real solutions, it never reaches the x-axis. The discriminant tells you which of the three pictures the graph makes, before you draw it.

Challenge

Medium

Work out the discriminant of x^2 + 6x + 9 = 0. Start from 6^2 - 4 × 1 × 9 and simplify it one step at a time. The sign of the final number tells you how many solutions the equation has.

Try it yourself

6^2 - 4 * 1 * 9

Simplify the expression