"One rational solution" $._("One rational solution") randRangeNonZero(-6, 6) * 2 randFromArray(getFactors(B * B / 4)) (B * B) / (4 * A) B * B - 4 * A * C

Describe the solutions to the following quadratic equation:

`expr(["+", ["*", A, ["^", "x", 2]], ["*", B, "x"], C]) = 0`

ANSWER

- One rational solution
- Two rational solutions
- Two irrational solutions
- One complex solution
- Two complex solutions

We could use the quadratic formula to solve for the solutions and see what they are, but there's a shortcut...

`\qquad x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}`

KhanUtil.processMath($("#quadratic")[0], "\\qquad x = \\dfrac{-b \\pm" + " \\sqrt{\\blue{b^2 - 4ac}}}{2a}", true);
Think about what the part of the quadratic formula under the radical tells us about the solutions.

Substitute the `a`, `b`, and `c` coefficients from the quadratic equation:

`\qquad\begin{array} && b^2-4ac \\ \\ =& (B)^2 - 4 ( A)(C) \\ \\ =& DISCRIMINANT \end{array}`

Because `\blue{b^2 - 4ac} = 0`, then the quadratic formula reduces to `\dfrac{-b}{2a}`, which means there is just one rational solution.

Because `\blue{b^2 - 4ac}` is negative, its square root is imaginary and the quadratic formula reduces to `\dfrac{-b \pm \sqrt{DISCRIMINANT}}{2a}` , which means there are two complex solutions.

Because `\blue{b^2 - 4ac}` is a perfect square, its square root is rational and the quadratic formula reduces to `\dfrac{-b \pm sqrt(DISCRIMINANT)}{2a}` , which means there are two rational solutions.

Because `\blue{b^2 - 4ac}` is not a perfect square, its square root is irrational and the quadratic formula reduces to `\dfrac{-b \pm \sqrt{DISCRIMINANT}}{2a}` , which means there are two irrational solutions.

"Two rational solutions" $._("Two rational solutions")
randRangeNonZero(-9, 9) randRangeNonZero(-9, 9) randRangeNonZero(-9, 9) B * B - 4 * A * C
"Two irrational solutions" $._("Two irrational solutions")
randRangeNonZero(-9, 9) randRangeNonZero(-9, 9) randRangeNonZero(-9, 9) B * B - 4 * A * C
"Two complex solutions" $._("Two complex solutions")
randRangeNonZero(-9, 9) randRangeNonZero(-9, 9) randRangeNonZero(-9, 9) B * B - 4 * A * C