Solving Quadratics With Complex Roots

We can use the discriminant of the given quadratic equation to determine the number and type of solutions. In the Quadratic Formula, b^2-4ac is the discriminant for any quadratic equation in standard form ax^2+bx+c=0. ax^2+bx+c=0 ⇕ x=- b±sqrt(b^2-4ac)/2a If we just want to know the number and type of solutions, not the solutions themselves, we only need to work with the discriminant. Let's first identify the values of a, b, and c in the given expression. 1x^2+ 11x+( - 7)=0 As we can see, a= 1, b= 11, and c= -7. We can now calculate the discriminant by substituting the values of a, b, and c into the discriminant expression.

Since the discriminant is 149, which is greater than zero, the quadratic equation has two real solutions.

Similar to Part A, we will start by identifying the values of a, b, and c of the given equation. 2x^2+( - 5)x+( 8)=0 In this equation, a= 2, b= -5, and c= 8. We can now substitute these values into the discriminant expression and then calculate it.

The discriminant is - 39, which is less than zero. If the discriminant of a quadratic equation is less than zero, then it has two imaginary solutions. Therefore, our quadratic equation has two imaginary solutions.

We will first write the given equation in the standard form of a quadratic equation by writing all the terms on the left-hand side. x^2-30x=- 225 ⇕ x^2-30x+225=0 Now that we rewrote the equation, we can identify the values of a, b, and c. 1x^2+( - 30)x+ 225=0 As we can see, a= 1, b= - 30, and c= 225. Finally, let's calculate the discriminant like we did in Parts A and B.

Since the discriminant is equal to zero, the quadratic equation has one real solution.

In the Quadratic Formula, b^2-4ac is the discriminant. ax^2+bx+c=0 ⇕ x=- b±sqrt(b^2-4ac)/2a Let's first rewrite the given quadratic equation in standard form. This will allow us to find the values for a, b, and c.

Now that the equation is rewritten in standard form, we can identify the values of a, b, and c. - 5 x^2+1=7x+12 ⇕ - 5x^2+( - 7)x+( - 11)=0 As we can see, a= - 5, b= - 7, and c= - 11. Now let's calculate the discriminant.

The discriminant is - 171.

We will use the Quadratic Formula to find the exact solutions of the given equation. x=- b±sqrt(b^2-4 a c)/2 a We have already identified the values of a, b, and c in Part A, as well as the value of the discriminant, b^2-4ac. a= - 5, b= - 7, c= - 11 Discriminant: - 171 Let's substitute these values into the Quadratic Formula.

We found by using the Quadratic Formula that the solutions of the given equation are x_1= - 7+ 3isqrt(19)10 and x_2= - 7- 3isqrt(19)10.

We are given a quadratic equation and one of its solutions. Equation: & Mt^2-12t+21=0 [0.3em] Solution: & x_1 = 2/3-1/3i It is given that M is a real number, and we can see that other coefficients are also real numbers. Recall that if a quadratic equation with real coefficients has a complex solution x_1=a+bi, then its conjugate is the other solution to the equation. x_2 = x_1 Based on this information, the second solution is the conjugate of x_1 = 23- 13i. To find the conjugate of a complex number, we need to change the sign of the imaginary part. x_2 &= x_1 &⇓ x_2 &= 2/3-1/3i &⇓ x_2 &= 2/3+1/3i Therefore, the second solution is x_2 = 23+ 13i.

We have two imaginary roots to the given quadratic equation. x_1=2/3-1/3i [0.5em] x_2=2/3+1/3i We want to add these two roots. Recall that to add or subtract two or more complex numbers, we combine its like terms — real parts with real parts and imaginary parts with imaginary parts.

The sum of x_1 and x_2 is 43.

We want to find the factors of the given algebraic expression and write its factored form. To do so, we will use the fact that the addition of two perfect squares can be written as the product of two complex conjugates. a^2+b^2 ⇔ (a+bi)(a-bi) We will first write each term as a perfect square. Then we will apply the formula.

The factored form of the given expression is (x+11i)(x-11i).

We want to find the solutions to the quadratic equation x^2+121=0. Note that we already factored the left-hand side of this equation in Part A. x^2+121=0 ⇕ (x+11i)(x-11i)=0 We can now solve this equation by using the Zero Product Property.

The x-values - 11i and 11i are solutions to the given equation.

We want to find the factors of the given algebraic expression and write its factored form. To do so, we will start by writing down the formulas to factor a quadratic equation that is a perfect square trinomial with an imaginary term. x^2 + 2bix - b^2=(x + bi)^2 [0.25em] x^2 - 2bix - b^2=(x - bi)^2 We want to factor the expression x^2-14ix-49. Since the coefficient of the x-term is negative, we will use the second formula. Let's rewrite the expression so that it resembles the formula.

Comparing the left-hand side expression to the second formula, we can conclude that b= 7. Finally, let's write the factored form of the expression. x^2-2( 7i)x- 7^2=(x- 7i)^2 The factored form of the given expression is (x-7i)^2.

We want to find the solutions to the given quadratic equation. We already factored the left-hand side of this equation in Part A. x^2-14ix-49=0 ⇕ (x-7i)^2=0 Looking at the factored form of the equation, we can conclude that the equation has only one solution, x=7i. x-7i=0 ⇔ x=7i

To find the price of the product if the monthly revenue is $ 4800, we can substitute R= 4800 into the function for the revenue of the company. R &= - p^2 + 72p - 400 & ⇓ 4800 &= - p^2 + 72p - 400 Let's rewrite the equation in standard form.

Now, we will evaluate the discriminant b^2-4ac. To do so, let's first identify the values of a, b, and c. - p^2 + 72p - 5200 = 0 ⇓ a= - 1, b= 72, c = - 5200 Now we can substitute these values into the expression representing the discriminant.

Since the discriminant is negative, the equation has no real solutions. In this context, it means that there is no price for which the monthly revenue of the company is $4800.

Although the given function has no real solutions, it actually does have solutions. The solutions are complex numbers, meaning that they do not make sense in a real-life context. To find these solutions, we can use the Quadratic Formula. p=- b±sqrt(b^2-4 a c)/2 a Note that we identified the values of a, b, and c in Part A, as well as the value of the discriminant b^2-4ac. This will make our job easier! a= - 1, b= 72, c = - 5200 [0.5em] Discriminant: - 15 616 Let's substitute these values into the Quadratic Formula.

The solutions to the equation are p_1 = 36 - 8isqrt(61) and p_2 = 36 + 8isqrt(61).