The Quadratic Formula and the Discriminant

In this lesson you will: Read a, b and c from a quadratic equation in standard form, signs included; Solve any quadratic equation with the quadratic formula; Use the discriminant b^2-4ac to predict the number of real solutions before solving.

One formula, every equation

Completing the square on ax^2+bx+c=0 in general produces a formula that solves every quadratic equation in one substitution. Write the equation in standard form, read off a, b and c, and evaluate.

The signs are the whole battle. In 2x^2-7x+3=0, b is -7, not 7, because the standard form is ax^2+bx+c with a plus in front of b. Write the three values down before you touch the formula.

The discriminant tells you what to expect

The expression under the radical, b^2-4ac, is called the discriminant. Call it D. Because it sits under a square root, its sign decides everything before you compute a single solution.

If D>0 there are two different real solutions. If D=0 the +/- adds and subtracts nothing, so there is exactly one. If D<0 there is no real square root to take, so the equation has no real solutions — in Algebra 1 that is a complete and correct answer.

What the three cases look like

The solutions of ax^2+bx+c=0 are the x-intercepts of y=ax^2+bx+c, so the discriminant is really counting crossings.

Two solutions means the curve cuts the x-axis twice. One solution means it touches and turns. No real solutions means it stays entirely above the axis, or entirely below it, and never arrives.

Common mistake — Sign errors under the radical

Solving 2x^2-7x+3=0, a student substitutes and writes b^2=-49, gets -49-24=-73, and concludes there are no real solutions.

The reasoning behind it is almost right. The student correctly noticed that b=-7, which is the step most people miss, and then carried the minus sign through the squaring as well.

But squaring a negative gives a positive: (-7)^2=49. The discriminant is 49-24=25, so sqrt(25)=5 and x=(7+/-5)/(4), giving x=3 and x=1/2.

The mirror-image slip is to write -b=-7 when b is already -7. Since b=-7, -b=+7, and the numerator starts with +7.

How to catch it: Look at the sign of b^2 in your working. Squaring destroys the sign, so that term is never negative. Write (-7)^2=49 on a line of its own before you subtract anything, and the error has nowhere to happen.