What is the discriminant and how many solutions does an equation have?

What it is and why it is enough

The discriminant is the middle coefficient squared, minus four times the leading coefficient times the constant. It is exactly what sits under the root in the formula.

Everything else in the formula is always defined. Only the root can fail, so the question of how many solutions there are reduces to a single question: what is the sign of this expression.

The three cases

Positive discriminant: the root is a positive number, and the plus-or-minus splits it into two distinct solutions.

Zero discriminant: the root is zero, so adding and subtracting it give the same number. There is one solution, usually called a double root.

Negative discriminant: no real number squares to a negative, so there is no real solution. The parabola in this case never touches the horizontal axis at all.

Questions with a parameter

A common question asks for the value of a parameter that gives exactly one solution. The translation is short: write the discriminant and set it equal to zero.

If two solutions are wanted, write an inequality greater than zero; if none, less than zero. In all three cases you solve an equation or inequality in the parameter, not in x.

Worked examples

1. How many solutions does x squared plus 4x plus 4 equals zero have?

   1. Coefficients: 1, 4 and 4
   2. Discriminant: 16 minus four times 1 times 4
   3. That is 16 minus 16, which is zero

   Answer: Exactly one. Indeed it is (x + 2) squared

2. How many solutions does x squared plus x plus 3 equals zero have?

   1. Coefficients: 1, 1 and 3
   2. Discriminant: 1 minus 12
   3. That is minus 11, which is negative

   Answer: No real solution. The parabola sits entirely above the axis

3. For which value of k does x squared plus kx plus 9 equals zero have one solution?

   1. One solution means a discriminant of zero
   2. Compute: k squared minus 36
   3. Set equal to zero: k squared equals 36
   4. Take the root, and both signs are possible

   Answer: k equals 6 or k equals minus 6

Common mistakes

Dropping the minus when the constant is negative

Minus times minus gives plus, so a negative constant makes the discriminant larger. Ignoring that turns an equation with two solutions into one that appears to have none.

Reading solution values from it

The discriminant says how many solutions there are, not what they are. For the values you carry on with the formula; it only tells you whether there is any point starting.

Calculating it before rearranging the equation

The coefficients are defined only in standard form, with zero on the other side. Calculating earlier uses numbers that are not the coefficients of the equation.

More in Quadratic Equations

• How do you expand brackets and what are the special products?
• How do you factorise an algebraic expression?
• How do you solve a quadratic equation by factorising?
• What is the quadratic formula and how do you use it?
• How do you find the vertex and the intercepts of a parabola?