Finding Discriminant and Number of Real Solutions

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True or false: for a quadratic function of form ax²+ bx + c = 0, if the discriminant b²- 4ac = 0, there is exactly one real root.

Explanation:

This is true. The discriminant b²- 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b²- 4ac and evaluate, three cases can happen:

b²- 4ac > 0

b²- 4ac = 0

b²- 4ac < 0

In the first case, having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes $\pm$ , this scenario yields two real results.

In the middle case (the case of our example), $\sqrt0 = 0$ . Going back to the quadratic formula $x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$ , you can see that when everything under the square root is simply 0, then you get only $x = \frac{-b }{2a}$ , which is why you have exactly one real root.

For the final case, if b²- 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

True or false: for a quadratic function of form ax²+ bx + c = 0, if the discriminant b²- 4ac > 0, there are exactly 2 distinct real roots of the equation.

Explanation:

b²- 4ac > 0

b²- 4ac = 0

b²- 4ac < 0

In the first case (the case of our example), having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes $\pm$ , this scenario yields two real results.

In the middle case, $\sqrt0 = 0$ . Going back to the quadratic formula $x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$ , you can see that when everything under the square root is simply 0, then you get only $x = \frac{-b }{2a}$ , which is why you have exactly one real root.

True or false: for a quadratic function of form ax²+ bx + c = 0, if the discriminant b²- 4ac < 0, there are exactly two distinct real roots.

Explanation:

This is false. The discriminant b²- 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b²- 4ac and evaluate, three cases can happen:

b²- 4ac > 0

b²- 4ac = 0

b²- 4ac < 0

For the final case (the case of our example), if b²- 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

Use the formula b² - 4ac to find the discriminant of the following equation: 4x² + 19x - 5 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 0

One real root: $\frac{-19}{8}$

Discriminant: 441

Two real roots: $x=\frac{1}{4}$ or x=-5

Discriminant: 281

Two imaginary roots: $\frac{-19\pm 281i}{8}$

Discriminant: 281

Two imaginary roots: $x = \frac{19\pm 281i}{8}$

Discriminant: 441

Two real roots: $x=-\frac{1}{4}$ or x=5

Correct answer:

Discriminant: 441

Two real roots: $x=\frac{1}{4}$ or x=-5

Explanation:

In the above equation, a = 4, b = 19, and c = -5. Therefore:

b² - 4ac = (19)² - 4(4)(-5) = 361 + 80 = 441.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two real roots.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-19\pm \sqrt{19^2 - 4\cdot 4\cdot -5}}{2\cdot 4}$

This simplifies to:

$x = \frac{-19\pm \sqrt{441}}{8}$

which simplifies to

$x = \frac{-19\pm21}{8}$

which gives us two answers:

$x=\frac{1}{4}$ or x=-5

These values of x are the two distinct real roots of the given equation.

Use the formula b² - 4ac to find the discriminant of the following equation: 4x² + 12x + 10 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots: $x=\frac{-3\pm i}{2}$

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots: $x=\frac{3\pm i}{2}$

Discriminant: 304

Types of Roots: Two distinct real roots

Exact Roots: $\frac{-3\pm \sqrt{19}}{2}$

Discriminant: 304

Types of Roots: Two distinct real roots

Exact Roots: $\frac{3\pm \sqrt{19}}{2}$

Discriminant: 16

Types of Roots: Two distinct real roots

Exact Roots: -1, -2

Correct answer:

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots: $x=\frac{-3\pm i}{2}$

Explanation:

In the above equation, a = 4, b = 12, and c = 10. Therefore:

b² - 4ac = (12)² - 4(4)(10) = 144 - 160 = -16.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-12\pm \sqrt{144 - 4\cdot 4\cdot 10}}{2\cdot 4}$

This simplifies to:

$x = \frac{-12\pm \sqrt{-16}}{8}$

$x=\frac{-12\pm 4i}{8}$

$x=\frac{-3\pm i}{2}$

In other words, our two distinct imaginary roots are $\frac{-3+i}{2}$ and $\frac{-3-i}{2}$

Use the formula b² - 4ac to find the discriminant of the following equation: -3x² + 6x - 3 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 0

One real root: x = -1

Discriminant: 72

Two distinct real roots: $x=1\pm \sqrt{2}$

Discriminant: 72

Two distinct real roots: $x=-1\pm \sqrt{2}$

Discriminant: 0

One real root: x = 1

Discriminant: -72

Two distinct imaginary roots: $x=1\pm 2i$

Correct answer:

Discriminant: 0

One real root: x = 1

Explanation:

In the above equation, a = -3, b = 6, and c = -3. Therefore:

b² - 4ac = (6)² - 4(-3)(-3) = 36 - 36 = 0.

Finally, we use the quadratic function to find these exact root. The quadratic formula is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-6\pm \sqrt{36 - 4\cdot -3\cdot -3}}{2\cdot -3}$

This simplifies to:

$x = \frac{-6\pm \sqrt{36 - 36}}{-6}$

which simplifies to

$x= \frac{-6\pm0}{-6}$

which gives us one answer: x = 1

This value of x is the one distinct real root of the given equation.

Use the formula b² - 4ac to find the discriminant of the following equation: x² + 5x + 4 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 9

Two real roots: x = -1 or x = -4

Discriminant: 0

One real root: $x=\frac{-5}{2}$

Discriminant: 9

Two real roots: x = 1 or x = 4

Discriminant: 41

Two imaginary roots: $x=\frac{5\pm 41i}{2}$

Discriminant: 41

Two imaginary roots: $x=\frac{-5\pm 41i}{2}$

Correct answer:

Discriminant: 9

Two real roots: x = -1 or x = -4

Explanation:

In the above equation, a = 1, b = 5, and c = 4. Therefore:

b² - 4ac = (5)² - 4(1)(4) = 25 - 16 = 9.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-5\pm \sqrt{5^2 - 4\cdot 1\cdot 4}}{2\cdot 1}$

This simplifies to:

$x = \frac{-5\pm \sqrt{9}}{2}$

which simplifies to

$x=\frac{-5\pm3}{2}$

which gives us two answers:

x = -1 or x = -4

These values of x are the two distinct real roots of the given equation.

Use the formula b² - 4ac to find the discriminant of the following equation: -x² + 3x - 3 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 0

One real root: $x=\frac{3}{2}$

Discriminant: -21

Two imaginary roots: $x=\frac{-3\pm\sqrt{21}i}{2}$

Discriminant: -8

Two imaginary roots: $x=\frac{3\pm\sqrt{3}i}{2}$ .

Discriminant: -21

Two imaginary roots: $x=\frac{3\pm\sqrt{21}i}{2}$

Discriminant: -8

Two imaginary roots: $x=\frac{-3\pm\sqrt{3}i}{2}$

Correct answer:

Discriminant: -8

Two imaginary roots: $x=\frac{3\pm\sqrt{3}i}{2}$ .

Explanation:

In the above equation, a = -1, b = 3, and c = -3. Therefore:

b² - 4ac = (3)² - 4(-1)(-3) = 9 - 12 = -3.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-3\pm \sqrt{3^2 - 4\cdot -1\cdot -3}}{2\cdot -1}$

This simplifies to:

$x = \frac{-3\pm \sqrt{-3}}{-2}$

Because $\sqrt{-3}=\sqrt{(-1)(3)}=\sqrt{3}i$ , this simplifies to $x=\frac{3\pm\sqrt{3}i}{2}$ . In other words, our two distinct imaginary roots are $x=\frac{3+\sqrt{3}i}{2}$ and $x=\frac{3-\sqrt{3}i}{2}$

Use the formula b² - 4ac to find the discriminant of the following equation: x² + 2x + 10 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 0

One real root: x = -1

Discriminant: -36

Two imaginary roots: $-1\pm3i$

Discriminant: -36

Two imaginary roots: $1\pm3i$

Discriminant: 36

Two real roots: x = -5 or x = 7

Discriminant: 36

Two real roots: x = 5 or x = -7

Correct answer:

Discriminant: -36

Two imaginary roots: $-1\pm3i$

Explanation:

In the above equation, a = 1, b = 2, and c = 10. Therefore:

b² - 4ac = (2)² - 4(1)(10) = 4 - 40 = -36.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-2\pm \sqrt{2^2 - 4\cdot 1\cdot 10}}{2\cdot 1}$

This simplifies to:

$x = \frac{-2\pm \sqrt{-36}}{2}$

Because $\sqrt{-36}=\sqrt{(-1)(36)}=6i$ , this simplifies to $x=\frac{-2\pm6i}{2}$ . We can further simplify this to $-1\pm3i$ . In other words, our two distinct imaginary roots are -1+3i and -1-3i .

Use the formula b² - 4ac to find the discriminant of the following equation: x² + 8x + 16 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 72

Two distinct real roots: $x=4\pm4\sqrt2$

Discriminant: 0

One real root: x = -4

Discriminant: 128

Two distinct real roots: $x=-4\pm4\sqrt{2}$

Discriminant: 0

One real root: x = 4

Discriminant: 0

One real root: x = 0

Correct answer:

Discriminant: 0

One real root: x = -4

Explanation:

In the above equation, a = 1, b = 8, and c = 16. Therefore:

b² - 4ac = (8)² - 4(1)(16) = 64 - 64 = 0.

Finally, we use the quadratic function to find these exact root. The quadratic formula is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-8\pm \sqrt{8^2 - 4\cdot 1\cdot 16}}{2\cdot 1}$

This simplifies to:

$x = \frac{-8\pm \sqrt{0}}{2}$

which simplifies to

x=-4

This value of x is the one distinct real root of the given equation.

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Finding Discriminant and Number of Real Solutions

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Finding Discriminant and Number of Real Solutions

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