Quadratics Reference
Discriminant and Quadratic Formula
The discriminant is the part of the quadratic formula that tells us how many real x-intercepts a quadratic has and whether factoring is likely to work cleanly.
Fact Table
| Result $D$ | What It Means for the Graph | Can You Factor It? |
|---|---|---|
| Positive square: $1,\ 4,\ 9,\ldots$ | Hits the x-axis twice. | Yes. Usually easy factoring. |
| Positive non-square: $2,\ 5,\ldots$ | Hits the x-axis twice. | No. Use the quadratic formula. |
| Zero: $0$ | Hits the x-axis exactly once. | Yes. Perfect square. |
| Negative: $-3,\ -10,\ldots$ | Never hits the x-axis. | No. Prime over the real numbers. |
Content Formulas
Discriminant
$$D=b^2-4ac$$
Quadratic Formula
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Use the discriminant before solving when the question asks about x-intercepts, real solutions, or whether a quadratic can factor cleanly.
Classic Examples
Positive Square
For $x^2-5x+6=0$, decide what the graph does and solve.
$$D=(-5)^2-4(1)(6)$$
$$D=25-24$$
$$D=1$$
$$(x-2)(x-3)=0$$
$$x=2,\quad x=3$$
Positive Non-Square
For $x^2-4x+1=0$, decide what the graph does and solve.
$$D=(-4)^2-4(1)(1)$$
$$D=16-4$$
$$D=12$$
$$x=\frac{4\pm\sqrt{12}}{2}$$
$$x=2\pm\sqrt3$$
Zero
For $x^2-6x+9=0$, decide what the graph does and solve.
$$D=(-6)^2-4(1)(9)$$
$$D=36-36$$
$$D=0$$
$$(x-3)^2=0$$
$$x=3$$
Negative
For $x^2+4x+8=0$, decide what the graph does.
$$D=4^2-4(1)(8)$$
$$D=16-32$$
$$D=-16$$
$$\text{No real x-intercepts.}$$
Extraneous Solution Check
Quadratic equations usually do not create extraneous solutions by themselves. Extraneous solutions appear when the original problem includes a square root, rational denominator, or another restriction.
Square Root Equation
Solve $\sqrt{x+5}=x-1$.
$$x+5=(x-1)^2$$
$$x+5=x^2-2x+1$$
$$0=x^2-3x-4$$
$$(x-4)(x+1)=0$$
$$x=4,\quad x=-1$$
Test both candidates in the original equation because squaring can create an extraneous solution.
$$\sqrt{-1+5}\ne -1-1$$
$$x=4$$