I have a few problems I need help with and also do have multiple choice. If I can have an explanation of how to get the answer that would be great.
1. Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots.
x^2-4x=5
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.
2.Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots.
-3x^3+19x-20=0
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.
3. Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots.
-5x^2-7x-5=0
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.
4. Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots
x^2-12=7x
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.
In google type:
quadratic equation online
When you see list of result click on:
Free Online Quadratic Equation Solver:Solve by Quadratic Formula
When page be open in rectangle type your equation
and cilck option:
solve it!
You will see solutions step-by step and discriminant discusion.
I am only going to do one,
1. Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots.
x^2-4x=5
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.
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Write with a zero on the right in form
a x^2 + b x + c = 0
1 x^2 -4 x - 5 = 0
b^2 = (-4)(-4) = 16
-4ac = -4(1)(-5) = +20
so
b^2 -4ac = 16+20 = 36
sqrt (b^2 - 4ac) = +6
THAT means:
x = [ 4 +/- 6 ] /2
Two REAL Rational roots
BECAUSE
sqrt (+36) is a real number
sqrt (+36) is not sqrt( 0) which would give one repeated root
sqrt(+36) is not sqrt(-36) which would leave you with a +/- 6i which is COMPLEX number
it is not sqrt (2) or something which is not a nice rational number and can not be expressed as a ratio of whole numebers
To determine the nature of the roots of a quadratic equation using the discriminant, you need to calculate the discriminant value, which is given by the formula b^2 - 4ac. Here, a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
Based on the value of the discriminant, you can determine the nature of the roots as follows:
1. For Question 1:
Equation: x^2 - 4x = 5
Comparing it with the general quadratic equation form ax^2 + bx + c = 0, you have a = 1, b = -4, and c = -5.
Now, calculate the discriminant: D = (-4)^2 - 4(1)(-5) = 16 + 20 = 36.
Since the discriminant is positive (greater than 0), the equation has two rational roots. Therefore, the answer is a.) Two rational roots.
2. For Question 2:
Equation: -3x^3 + 19x - 20 = 0
Comparing it with the general quadratic equation form ax^2 + bx + c = 0, you have a = -3, b = 19, and c = -20.
Now, calculate the discriminant: D = (19)^2 - 4(-3)(-20) = 361 - 240 = 121.
Since the discriminant is positive (greater than 0), the equation has two rational roots. Therefore, the answer is a.) Two rational roots.
3. For Question 3:
Equation: -5x^2 - 7x - 5 = 0
Comparing it with the general quadratic equation form ax^2 + bx + c = 0, you have a = -5, b = -7, and c = -5.
Now, calculate the discriminant: D = (-7)^2 - 4(-5)(-5) = 49 - 100 = -51.
Since the discriminant is negative (less than 0), the equation has two complex roots. Therefore, the answer is c.) Two complex roots.
4. For Question 4:
Equation: x^2 - 12 = 7x
Rearrange to get it into general quadratic equation form: x^2 - 7x - 12 = 0
Comparing it with the general quadratic equation form ax^2 + bx + c = 0, you have a = 1, b = -7, and c = -12.
Now, calculate the discriminant: D = (-7)^2 - 4(1)(-12) = 49 + 48 = 97.
Since the discriminant is positive (greater than 0), the equation has two irrational roots. Therefore, the answer is d.) Two irrational roots.
Remember, the discriminant helps determine the nature of the roots by considering whether it is positive, negative, or zero.