How To Find How Many Roots An Equation Has

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True or simulated: for a quadratic function of course ax²+ bx + c = 0, if the discriminant b²- 4ac = 0, there is exactly ane existent root.

Caption:

This is truthful. The discriminant b²- 4ac is the part of the quadratic formula that lives within of a square root function. As you plug in the constants a, b, and c into bⁱⁱ- 4ac and evaluate, 3 cases can happen:

b²- 4ac > 0

b^two- 4ac = 0

b²- 4ac < 0

In the outset case, having a positive number under a square root role will yield a result that is a positive number reply. Even so, considering the quadratic part includes $\pm$ , this scenario yields two real results.

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

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

True or false: for a quadratic part of grade ax²+ bx + c = 0, if the discriminant bⁱⁱ- 4ac > 0, at that place are exactly 2 distinct real roots of the equation.

Caption:

This is truthful. The discriminant b^two- 4ac is the part of the quadratic formula that lives within of a square root part. As you plug in the constants a, b, and c into b²- 4ac and evaluate, iii cases tin happen:

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 part will yield a event that is a positive number reply. However, because the quadratic function includes $\pm$ , this scenario yields two real results.

In the middle case, $\sqrt0 = 0$ . Going dorsum to the quadratic formula $x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$ , y'all can see that when everything under the square root is merely 0, then you get only $x = \frac{-b }{2a}$ , which is why you have exactly one existent 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; notwithstanding, you will have exactly 2 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 two distinct existent roots.

Explanation:

This is false. The discriminant bⁱⁱ- 4ac is the part of the quadratic formula that lives within of a foursquare root function. Every bit you plug in the constants a, b, and c into b²- 4ac and evaluate, iii cases tin can happen:

b^two- 4ac > 0

b²- 4ac = 0

b²- 4ac < 0

In the kickoff instance, having a positive number under a square root office will yield a event that is a positive number answer. However, because the quadratic role includes $\pm$ , this scenario yields 2 real results.

In the centre case, $\sqrt0 = 0$ . Going back to the quadratic formula $x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$ , you tin see that when everything nether the square root is simply 0, then you become only $x = \frac{-b }{2a}$ , which is why you have exactly i existent root.

For the final example (the case of our example), if b²- 4ac < 0, that means you have a negative number under a square root. This means that you volition non accept any real roots of the equation; still, you will have exactly two imaginary roots of the equation.

Use the formula bⁱⁱ - 4ac to observe the discriminant of the following equation: 4x² + 19x - five = 0.

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

Possible Answers:

Discriminant: 441

2 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: 0

I existent root: $\frac{-19}{8}$

Discriminant: 441

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

Correct answer:

Discriminant: 441

2 existent roots: $x=\frac{1}{4}$ or x=-5

Caption:

In the to a higher place equation, a = 4, b = xix, and c = -v. Therefore:

b² - 4ac = (xix)² - iv(four)(-5) = 361 + 80 = 441.

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

Finally, we use the quadratic role to notice these verbal roots. The quadratic part is:

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

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

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

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 bⁱⁱ - 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 office to find the exact roots of the equation.

Possible Answers:

Discriminant: 304

Types of Roots: Ii singled-out 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 existent roots; 2 distinct imaginary roots

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

Discriminant: 16

Types of Roots: 2 distinct real roots

Exact Roots: -1, -2

Discriminant: -16

Types of Roots: No real roots; two singled-out imaginary roots

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

Correct answer:

Discriminant: -xvi

Types of Roots: No existent roots; 2 singled-out imaginary roots

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

Explanation:

In the to a higher place equation, a = 4, b = 12, and c = 10. Therefore:

b^two - 4ac = (12)² - four(4)(10) = 144 - 160 = -sixteen.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, in that location is exactly ane real root. When the discriminant is less than zero, there are no existent roots, simply there are exactly two distinct imaginary roots. In this case, we have two singled-out imaginary roots.

Finally, we apply the quadratic function to discover these exact roots. The quadratic office is:

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

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

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

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

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

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

Possible Answers:

Discriminant: 72

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

Discriminant: -72

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

Discriminant: 0

One real root: x = -i

Discriminant: 0

One real root: x = 1

Discriminant: 72

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

Correct respond:

Discriminant: 0

One real root: x = 1

Explanation:

In the above equation, a = -iii, b = six, and c = -three. Therefore:

b² - 4ac = (half-dozen)² - 4(-three)(-3) = 36 - 36 = 0.

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 in that location are exactly two distinct imaginary roots. In this example, at that place is exactly ane real root.

Finally, nosotros 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 united states of america one respond: x = 1

This value of x is the 1 distinct real root of the given equation.

Apply the formula b² - 4ac to discover the discriminant of the following equation: x² + 5x + iv = 0.

And so state how many roots it has, and whether they are real or imaginary. Finally, apply the quadratic part to discover the exact roots of the equation.

Possible Answers:

Discriminant: 41

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

Discriminant: nine

Two existent roots: x = 1 or x = 4

Discriminant: 0

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

Discriminant: 9

Two real roots: x = -1 or ten = -four

Discriminant: 41

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

Correct respond:

Discriminant: ix

Two existent roots: x = -1 or x = -iv

Explanation:

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

b² - 4ac = (v)ⁱⁱ - four(1)(4) = 25 - 16 = 9.

When the discriminant is greater than 0, there are two singled-out real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zip, there are no real roots, but at that place are exactly two singled-out imaginary roots. In this example, we have two real roots.

Finally, we use the quadratic role to find these verbal 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 ii answers:

x = -one or ten = -four

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

Use the formula bⁱⁱ - 4ac to find the discriminant of the following equation: -ten² + 3x - 3 = 0.

Then country how many roots it has, and whether they are real or imaginary. Finally, employ the quadratic office to find the exact roots of the equation.

Possible Answers:

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

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: -viii

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

Explanation:

In the in a higher place equation, a = -one, b = iii, and c = -3. Therefore:

b^two - 4ac = (three)² - four(-1)(-iii) = 9 - 12 = -iii.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly 1 existent root. When the discriminant is less than nix, there are no real roots, but there are exactly 2 singled-out imaginary roots. In this instance, nosotros have 2 singled-out imaginary roots.

Finally, we utilise the quadratic function to find these verbal roots. The quadratic office 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 ii 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 country how many roots information technology has, and whether they are real or imaginary. Finally, utilise the quadratic function to notice the verbal roots of the equation.

Possible Answers:

Discriminant: 36

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

Discriminant: 36

Two real roots: ten = 5 or x = -seven

Discriminant: -36

Two imaginary roots: $1\pm3i$

Discriminant: -36

2 imaginary roots: $-1\pm3i$

Discriminant: 0

One real root: x = -1

Correct answer:

Discriminant: -36

2 imaginary roots: $-1\pm3i$

Caption:

In the to a higher place equation, a = ane, b = 2, and c = 10. Therefore:

b² - 4ac = (2)² - 4(one)(ten) = four - 40 = -36.

When the discriminant is greater than 0, in that location are ii distinct real roots. When the discriminant is equal to 0, there is exactly 1 existent root. When the discriminant is less than zip, there are no real roots, only there are exactly two singled-out imaginary roots. In this case, we have ii distinct imaginary roots.

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

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

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

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

This simplifies to:

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

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

Employ the formula b² - 4ac to find the discriminant of the post-obit equation: x² + 8x + xvi = 0.

So state how many roots information technology has, and whether they are real or imaginary. Finally, use the quadratic function to detect the exact roots of the equation.

Possible Answers:

Discriminant: 0

One existent root: x = -4

Discriminant: 0

I existent root: x = 0

Discriminant: 72

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

Discriminant: 0

1 real root: ten = iv

Discriminant: 128

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

Correct answer:

Discriminant: 0

One existent root: x = -4

Explanation:

In the higher up equation, a = 1, b = eight, and c = 16. Therefore:

b² - 4ac = (8)ⁱⁱ - iv(1)(sixteen) = 64 - 64 = 0.

When the discriminant is greater than 0, there are 2 distinct existent roots. When the discriminant is equal to 0, at that place is exactly one real root. When the discriminant is less than zero, there are no real roots, just there are exactly two distinct imaginary roots. In this example, there is exactly one real root.

Finally, we use the quadratic function to notice 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 go:

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

This simplifies to:

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

which simplifies to

x=-4

This value of x is the i singled-out real root of the given equation.

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