Part 2 of homework set.
6. Solve the quadratic using the quadratic formula.
2x^2+5x+9=0
7. Find the discriminant. Then determine the number and types of solutions.
2x^2+3x-18=0
8. Simplify
Root -27
9. Simplify 8/(3+2i)
#6. Surely by now you can do this...
#7. b^2-4ac = 9+4*2*18
#8. 3√-3 = 3√3 i
#9. Multiply top and bottom by conjugate:
8(3-2i)/13
Again thanks.
6. To solve the quadratic equation using the quadratic formula, we first need to identify the coefficients of the equation in the standard form, ax^2 + bx + c = 0. In this case, a = 2, b = 5, and c = 9.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac))/(2a)
Substituting the given coefficients into the formula:
x = (-5 ± √(5^2 - 4*2*9))/(2*2)
= (-5 ± √(25 - 72))/(4)
= (-5 ± √(-47))/(4)
Since we have a negative value (under the square root), we know that the quadratic equation does not have real solutions. The solutions are complex.
7. The discriminant, denoted by Δ, is given by the formula:
Δ = b^2 - 4ac
Substituting the coefficients from the equation 2x^2 + 3x - 18 = 0, we get:
Δ = (3^2) - 4(2)(-18)
= 9 + 144
= 153
The discriminant is 153.
Next, we can determine the number and types of solutions based on the discriminant.
If Δ > 0, then the quadratic equation has two distinct real solutions.
If Δ = 0, then the quadratic equation has one real solution (also called a repeated root).
If Δ < 0, then the quadratic equation has two complex solutions.
In this case, Δ = 153, which is greater than zero. Therefore, the quadratic equation 2x^2 + 3x - 18 = 0 has two distinct real solutions.
8. To simplify the expression √(-27), we can rewrite it as √(9 * -3).
Since we know that the square root of 9 is 3, we can simplify it further:
√(-27) = √(9 * -3)
= √9 * √(-3)
= 3i√3
Therefore, the simplification of √(-27) is 3i√3, where "i" represents the imaginary unit.
9. To simplify the expression 8/(3+2i), we need to rationalize the denominator.
To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which is 3-2i.
8/(3+2i) * (3-2i)/(3-2i)
= (8 * (3-2i))/(9-4i+6i-4i^2)
= (24 - 16i)/(9+4)
= (24 - 16i)/13
Therefore, the simplified form of 8/(3+2i) is (24 - 16i)/13.
6. To solve the quadratic equation using the quadratic formula, we have the equation 2x^2 + 5x + 9 = 0. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In this formula, a, b, and c represent the coefficients of the quadratic equation. Let's substitute the values into the formula:
a = 2, b = 5, c = 9
x = (-5 ± √(5^2 - 4 * 2 * 9)) / (2 * 2)
Simplifying further, we get:
x = (-5 ± √(25 - 72)) / 4
x = (-5 ± √(-47)) / 4
Since the discriminant (√(-47)) is imaginary, the quadratic has complex solutions. In this case, we cannot find real roots for the equation 2x^2 + 5x + 9 = 0.
7. The discriminant (denoted as Δ) can be calculated using the formula:
Δ = b^2 - 4ac
In the equation 2x^2 + 3x - 18 = 0, the coefficients are a = 2, b = 3, and c = -18. Let's plug these values into the formula:
Δ = (3^2) - 4 * 2 * (-18)
Simplifying further, we get:
Δ = 9 + 144
Δ = 153
The discriminant is 153.
Next, we can determine the number and types of solutions based on the discriminant:
- If the discriminant Δ > 0, there are two distinct real solutions.
- If the discriminant Δ = 0, there is one real solution (a repeated root).
- If the discriminant Δ < 0, there are no real solutions, only complex solutions.
In our case, Δ = 153, which is greater than 0, so there are two distinct real solutions for the equation 2x^2 + 3x - 18 = 0.
8. To simplify the square root of -27, we can factor out -1 from the square root:
√(-27) = √(9 * -3) = √(9) * √(-3) = 3i√3
Therefore, the simplified form of √(-27) is 3i√3.
9. To simplify 8/(3 + 2i), we need to rationalize the denominator. We can do this by multiplying both the numerator and denominator by the conjugate of 3 + 2i, which is 3 - 2i:
8/(3 + 2i) * (3 - 2i)/(3 - 2i)
Expanding the numerator and denominator, we get:
(8 * 3 - 8 * 2i + 8 * 3i - 8 * -2i) / (9 - 4i^2)
Simplifying further, we get:
(24 - 16i + 24i + 16) / (9 + 4)
(40 + 8i) / 13
Therefore, the simplified form of 8/(3 + 2i) is (40 + 8i) / 13.