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Precalculus : Find roots of quadratic equation using discriminant

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Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

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.

Possible Answers:

True

False

Correct answer:

True

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.

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Example Question #2 : Find Roots Of Quadratic Equation Using Discriminant

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.

Possible Answers:

True

False

Correct answer:

True

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.

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Example Question #3 : Find Roots Of Quadratic Equation Using Discriminant

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.

Possible Answers:

True

False

Correct answer:

False

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.

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Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

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: 281

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

Discriminant: 441

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

Discriminant: 441

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

Discriminant: 281

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

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.

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Example Question #2 : Find Roots Of Quadratic Equation Using Discriminant

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: 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: 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: 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 $\frac{-3-i}{2}$

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Example Question #4 : Find Roots Of Quadratic Equation Using Discriminant

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: 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: -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.

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Example Question #3 : Find Roots Of Quadratic Equation Using Discriminant

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: 41

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

Discriminant: 0

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

Discriminant: 9

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

Discriminant: 9

Two real roots: x = 1 or x = 4

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.

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Example Question #4 : Find Roots Of Quadratic Equation Using Discriminant

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: -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: 0

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

Discriminant: -8

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

Discriminant: -21

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

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

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Example Question #5 : Find Roots Of Quadratic Equation Using Discriminant

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: 36

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

Discriminant: -36

Two imaginary roots: $-1\pm3i$

Discriminant: 0

One real root: x = -1

Discriminant: -36

Two imaginary roots: $1\pm3i$

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 .

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Example Question #6 : Find Roots Of Quadratic Equation Using Discriminant

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: 0

One real root: x = 0

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: 72

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

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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Precalculus : Find roots of quadratic equation using discriminant

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #2 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #3 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #2 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #4 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #3 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #4 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #5 : Find Roots Of Quadratic Equation Using Discriminant

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