Algebra II - Discriminants

Determine the discriminant:

$\textup{Cannot be determined.}$

Explanation

The discriminant is the term inside the square root of the quadratic equation.

The polynomial is provided in standard form .

Substitute the variables into the equation.

The answer is:

The equation

$4x ^{2} + Bx + 27= 0$

has two imaginary solutions.

For what positive integer values of is this possible?

$\left { 1, 2, 3,...20\right }$

$\left { 21, 22, 23, ... \right }$

$\left { 1, 2, 3,...11\right }$

$\left { 12, 13, 14,...\right }$

All positive integers

Explanation

For the equation

$ax^{2} + bx + c = 0$

to have two imaginary solutions, its discriminant $b^{2} - 4ac$ must be negative. Set and solve for in the inequality

$b^{2} - 4ac < 0$

$B^{2} - 4 (4)(27)< 0$

$B^{2} - 432< 0$

$B^{2} < 432$

$\left | B \right | < \sqrt{432} \approx 20.8$

Therefore, if is a positive integer, it must be in the set $\left { 1, 2, 3,...20\right }$ .

The equation

$3x ^{2} + Bx + 28= 0$

has two real solutions.

For what positive integer values of is this possible?

$\left { 19, 20, 21, ...\right }$

$\left { 5, 6, 7, ...\right }$

$\left { 1,2, 3,4\right }$

$\left { 1,2, 3,...,18 \right }$

All positive integers

Explanation

For the equation

$ax^{2} + bx + c = 0$

to have two real solutions, its discriminant $b^{2} - 4ac$ must be positive. Set and solve for in the inequality

$b^{2} - 4ac > 0$

$B^{2} - 4 (3)(28)> 0$

$B^{2} - 336> 0$

$B^{2} > 336$

$\left | B\right | > \sqrt{336} \approx 18.3$

Therefore, if is a positive integer, it must be in the set $\left { 19, 20, 21, ...\right }$

Determine the discriminant of:

Explanation

The equation is given in the form of .

Write the formula for the discriminant.

Identify the coefficients.

Substitute the values into the equation.

The answer is:

Given , what is the value of the discriminant?

$^{\sqrt{21}}$

Explanation

The correct answer is . The discriminant is equal to portion of the quadratic formula. In this case, "" corresponds to the coefficient of , "" corresponds to the coefficient of , and "" corresponds to . So, the answer is , which is equal to .

Determine the discriminant of the parabola:

Explanation

Write the formula for the discriminant. This is the term inside the radical of the quadratic equation.

Substitute the values into the equation.

The answer is:

Evaluate the discriminant, if any:

$\frac{5}{3}$

$\textup{The discriminant cannot be determined.}$

Explanation

The formula to determine the discriminant is:

To determine the discriminant, we will need to put the equation in standard form:

Add on both sides.

Subtract on both sides.

Reorder the terms on the left.

Divide by two on both sides.

$\frac{x^2-2x-6 }{2}= \frac{2y}{2}$

The equation in standard form is: $y=\frac{1}{2}x^2-x-3$

The coefficients can be determined to calculate the discriminant.

$a=\frac{1}{2}, b=-1, c=-3$

Substitute the values into the formula.

$D=2b^2-4ac = (-1)^2-4(\frac{1}{2})(-3) = 1+6 = 7$

The answer is:

Use the discriminant to determine the number of real roots the function has:

The function has two real roots

The function has one real root

The function has no real roots

It is impossible to determine

Explanation

Using the discriminant, which for a polynomial

is equal to

we can determine the number of roots a polynomial has. If the discriminant is positive, then the polynomial has two real roots. If it is equal to zero, the polynomial has one real root. If it is negative, then the polynomial has two roots which are complex conjugates of one another.

For our function, we have