The Discriminant

Show your work for each problem.

1.
What is the discriminant formula for $ax^2 + bx + c = 0$ ?
1. a) $b^2 - 4ac$
2. b) $b^2 + 4ac$
3. c) $b - 4ac$
4. d) $2b - 4ac$
Answer: $b^2 - 4ac$
The discriminant is $\Delta = b^2 - 4ac$ . It's the part under the square root in the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
2.
If the discriminant is positive ( $\Delta > 0$ ), how many real solutions does the equation have?
1. a)Three real solutions
2. b)Two distinct real solutions
3. c)One real solution
4. d)No real solutions
Answer: Two distinct real solutions
When $\Delta > 0$ , we can take the square root of a positive number, giving us two different values with $\pm\sqrt{\Delta}$ , resulting in two distinct real solutions.
3.
Calculate the discriminant for $x^2 + 4x + 4 = 0$ .
Answer: 0
$\Delta = 4^2 - 4(1)(4) = 16 - 16 = 0$ . Since $\Delta = 0$ , there is exactly one (repeated) solution: $x = -2$ .
4.
Calculate the discriminant for $x^2 - 7x + 10 = 0$ .
Answer: 9
$\Delta = (-7)^2 - 4(1)(10) = 49 - 40 = 9$ . Since $\Delta > 0$ , there are two real solutions.
5.
A quadratic equation has discriminant $\Delta = -16$ . What does this tell us?
1. a)No real solutions
2. b)Four real solutions
3. c)One real solution
4. d)Two real solutions
Answer: No real solutions
When $\Delta < 0$ , we cannot take the square root of a negative number in the real number system. The equation has no real solutions (only complex solutions with $i$ ).
6.
Calculate the discriminant for $2x^2 + 5x - 3 = 0$ .
Answer: 49
$\Delta = 5^2 - 4(2)(-3) = 25 - (-24) = 25 + 24 = 49$ . Since $\Delta > 0$ , there are two real solutions.
7.
Calculate the discriminant of $3x^2 - 2x + 5 = 0$ and determine the nature of solutions.
Answer: $Δ = -56$ , no real solutions
1. What is the value of $a$ ? 3
2. What is the value of $b$ ? -2
3. What is the value of $c$ ? 5
4. Calculate $b^2$ 4
5. Calculate $4ac$ 60
6. What is the discriminant? -56
8.
The parabola $y = x^2 - 4x + 5$ has its vertex above the x-axis. What can you conclude about the discriminant?
1. a)Cannot determine
2. b) $\Delta > 0$
3. c) $\Delta < 0$
4. d) $\Delta = 0$
Answer: $\Delta < 0$
Since the parabola opens upward (a > 0) and has its vertex above the x-axis, it never crosses the x-axis. This means no real roots, so $\Delta < 0$ . Indeed, $\Delta = 16 - 20 = -4$ .
9.
Find the discriminant of $-x^2 + 6x - 9 = 0$ and interpret the result.
Answer: $Δ = 0$ , one repeated solution ( $x = 3$ )
1. What is $a$ ? -1
2. What is $b$ ? 6
3. What is $c$ ? -9
4. Calculate $b^2$ 36
5. Calculate $4ac$ 36
6. What is the discriminant? 0
10.
How many real solutions does $x^2 + 2x + 5 = 0$ have? (Answer: 0, 1, or 2)
Answer: 0
$\Delta = 4 - 20 = -16 < 0$ . Since the discriminant is negative, there are no real solutions (0 real solutions).
11.
For what value of $k$ does $x^2 - 6x + k = 0$ have exactly one solution?
Answer: 9
1. For exactly one solution, what must the discriminant equal? 0
2. Write the discriminant formula with $a=1$ , $b=-6$ , $c=k$ 36 - 4k
3. Set the discriminant equal to zero and solve: $36 - 4k = 0$ . What is $k$ ? 9
12.
For which values of $m$ does $x^2 + mx + 4 = 0$ have two distinct real solutions?
1. a)only $m > 4$
2. b) $m < -4$ or $m > 4$
3. c) $m < -16$ or $m > 16$
4. d) $-4 < m < 4$
Answer: $m < -4$ or $m > 4$
For two distinct solutions: $\Delta > 0$ . So $m^2 - 4(1)(4) > 0$ , giving $m^2 > 16$ . This means $m > 4$ or $m < -4$ : negative values work too, for example $m = -5$ gives $\Delta = 25 - 16 = 9 > 0$ .
13.
A ball is thrown upward with height $h = -5t^2 + 30t + 2$ meters. Use the discriminant to determine if it reaches 50 meters.
Answer: No: $Δ = -60 < 0$ , so the ball never reaches 50 meters
1. Set $h = 50$ and rearrange to standard form. What equation do you get? -5t^2 + 30t - 48 = 0
2. What is $c$ in $-5t^2 + 30t - 48 = 0$ ? -48
3. Calculate $b^2$ 900
4. Calculate $4ac$ 960
5. What is the discriminant? -60
6. Does the ball reach 50 meters? no
14.
The equation $2x^2 - 8x + k = 0$ has no real solutions. Which of these could be $k$ ?
1. a) $k = 4$
2. b) $k = 2$
3. c) $k = 10$
4. d) $k = 8$
Answer: $k = 10$
For no real solutions: $\Delta < 0$ . So $(-8)^2 - 4(2)(k) < 0$ , giving $64 - 8k < 0$ , therefore $k > 8$ . Only $k = 10$ satisfies this condition.
15.
The quadratic equation $px^2 + 6x + 1 = 0$ (with $p \neq 0$ ) has exactly one solution. What is $p$ ?
Answer: 9
1. For one solution, the discriminant must equal what? 0
2. Write the discriminant: $b^2 - 4ac$ with $a = p$ , $b = 6$ , $c = 1$ 36 - 4p
3. Set $36 - 4p = 0$ and solve for $p$ 9