Positive Discriminant: Two Solutions

Determine the nature of solutions for $x^2 - 5x + 6 = 0$

Identify a, b, and c: $a = 1$, $b = -5$, $c = 6$ = Coefficients identified

Calculate the discriminant: $\Delta = (-5)^2 - 4(1)(6) = 25 - 24$ = $\Delta = 1$

Interpret the result: Since $\Delta = 1 > 0$, there are two distinct real solutions = Two real solutions

Verify by solving: $x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}$, so $x = 3$ or $x = 2$ = $x = 2$ and $x = 3$ confirmed

Answer: Two distinct real solutions: $x = 2$ and $x = 3$

Zero Discriminant: One Solution

Determine the nature of solutions for $x^2 - 6x + 9 = 0$

Identify a, b, and c: $a = 1$, $b = -6$, $c = 9$ = Coefficients identified

Calculate the discriminant: $\Delta = (-6)^2 - 4(1)(9) = 36 - 36$ = $\Delta = 0$

Interpret the result: Since $\Delta = 0$, there is exactly one repeated real solution = One repeated solution

Verify by solving: $x = \frac{6 \pm \sqrt{0}}{2} = \frac{6}{2} = 3$ = $x = 3$ (double root)

Answer: One repeated real solution: $x = 3$ (the parabola touches the x-axis at exactly one point)

Negative Discriminant: No Real Solutions

Determine the nature of solutions for $2x^2 + x + 1 = 0$

Identify a, b, and c: $a = 2$, $b = 1$, $c = 1$ = Coefficients identified

Calculate the discriminant: $\Delta = (1)^2 - 4(2)(1) = 1 - 8$ = $\Delta = -7$

Interpret the result: Since $\Delta = -7 < 0$, there are no real solutions = No real solutions

Graphical meaning: The parabola opens upward and never crosses the x-axis = Parabola is entirely above x-axis

Answer: No real solutions (the parabola does not intersect the x-axis)

Finding Values for a Specific Number of Solutions

For what value of $k$ does $x^2 + kx + 9 = 0$ have exactly one solution?

Set up the condition: For one solution, $\Delta = 0$ = $b^2 - 4ac = 0$

Substitute values: $k^2 - 4(1)(9) = 0$ = $k^2 - 36 = 0$

Solve for k: $k^2 = 36$, so $k = \pm 6$ = $k = 6$ or $k = -6$

Verify one solution: With $k = 6$: $x^2 + 6x + 9 = (x+3)^2 = 0$ = $x = -3$ (one solution)

Answer: $k = 6$ or $k = -6$

Mistake: Forgetting the negative sign when $b$ is negative

Why: When $b = -5$, students calculate $b^2 = -25$ instead of $(-5)^2 = 25$

Correct: Always square the entire coefficient including its sign: $(-5)^2 = 25$

Mistake: Confusing $b^2 - 4ac$ with $b - 4ac$

Why: Students sometimes forget to square $b$, drastically changing the result

Correct: The discriminant is $b^2 - 4ac$. The $b$ MUST be squared.

Mistake: Thinking $\Delta = 0$ means no solutions

Why: Zero seems like nothing, so students assume no solutions exist

Correct: $\Delta = 0$ means exactly ONE solution (a repeated root), not zero solutions

Mistake: Misidentifying coefficients when equation is not in standard form

Why: Students use wrong values for $a$, $b$, $c$ if equation is like $3x + x^2 = 5$

Correct: Always rewrite in standard form $ax^2 + bx + c = 0$ first: $x^2 + 3x - 5 = 0$

Projectile Motion

When launching a projectile, the discriminant tells us whether it will reach a certain height.

A ball thrown upward follows $h = -5t^2 + 20t + 1$. To find when $h = 25$: $-5t^2 + 20t - 24 = 0$. The discriminant $\Delta = 400 - 480 = -80 < 0$ tells us the ball never reaches 25 meters.

Business Break-Even Analysis

Companies use the discriminant to determine if profit targets are achievable.

If profit $P = -2x^2 + 40x - 100$, finding when $P = 150$ gives $-2x^2 + 40x - 250 = 0$. With $\Delta = 1600 - 2000 = -400 < 0$, profit of 150 is impossible.

Q: Why is it called the discriminant?

A: The word discriminant comes from discriminate meaning to distinguish or tell apart. The discriminant discriminates between different types of solutions: two real, one real, or no real solutions.

Q: What happens to solutions when the discriminant is negative?

A: The solutions become complex numbers involving $i = \sqrt{-1}$. For example, if $\Delta = -4$, then $\sqrt{\Delta} = \sqrt{-4} = 2i$. Complex solutions are beyond most algebra courses but are studied in advanced math.

Q: Can the discriminant help with graphing?

A: Absolutely! The discriminant tells you how many x-intercepts the parabola has: $\Delta > 0$ means 2 x-intercepts, $\Delta = 0$ means 1 x-intercept (vertex touches x-axis), $\Delta < 0$ means no x-intercepts.