Problem 45 Use the quadratic formula to sol... [FREE SOLUTION]

Chapter 8: Problem 45

Use the quadratic formula to solve each equation. (All solutions for these equations are nonreal complex numbers.) $$ (6 x+1)(x-2)=(x+2)(x-5) $$

Short Answer

Expert verified

$x = \frac{4 \pm 2\sqrt{6}i}{5}$

Step by step solution

- Expand both sides of the equation

Expand the left-hand side: ewline(6x + 1)(x - 2) = 6x^2 - 12x + x - 2 = 6x^2 - 11x - 2ewlineExpand the right-hand side: ewline(x + 2)(x - 5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10

- Set the equation to zero

Combine all terms to one side of the equation to set it to zero: ewline6x^2 - 11x - 2 = x^2 - 3x - 10 ewline6x^2 - 11x - 2 - x^2 + 3x + 10 = 0 ewlineWhich simplifies to: 5x^2 - 8x + 8 = 0

- Identify coefficients for the quadratic formula

Identify the coefficients from the standard quadratic form x^2 + bx + c = 0.ewlinea = 5, b = -8, c = 8

- Apply the quadratic formula

Use the quadratic formula: ewline$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ewlineSubstitute the coefficients: ewline$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(5)(8)}}{2(5)}$ewlineSimplify inside the square root: ewline$x = \frac{8 \pm \sqrt{64 - 160}}{10}$ewline$x = \frac{8 \pm \sqrt{-96}}{10}$ewline$x = \frac{8 \pm 4\sqrt{6}i}{10}$ewlineWhich further simplifies to:ewline$x = \frac{4 \pm 2\sqrt{6}i}{5}$

- Write the solution in simplest form

The solutions are: ewline$x = \frac{4 + 2\sqrt{6}i}{5}$ ewlineand ewline$x = \frac{4 - 2\sqrt{6}i}{5}$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic formula

Quadratic equations of the form $ax^2 + bx + c = 0$ can be solved using the quadratic formula. The quadratic formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The symbol $ \pm $ means there can be two solutions: one with addition and one with subtraction.

Let's break down each part:

a, b, and c: These are the coefficients from the quadratic equation. Here, $a$ multiplies $x^2$, $b$ multiplies $x$, and $c$ is the constant term.

b虏 - 4ac: This part is called the discriminant. It determines the nature of the roots of the equation:

If the discriminant is positive, there are two real solutions.
If the discriminant is zero, there is one real solution.
If the discriminant is negative, there are two complex solutions.

The quadratic formula works for any quadratic equation, regardless of whether the solutions are real or complex. It's a powerful tool for solving quadratic equations.

complex numbers

In this problem, the solutions turned out to be complex numbers. Complex numbers are numbers that have a real part and an imaginary part. They are written in the form $a + bi$ where $i$ is the imaginary unit, defined as $i^2 = -1$.
For example, if you have $2 + 3i$, $2$ is the real part, and $3i$ is the imaginary part.

Key points about complex numbers:

Complex numbers allow us to solve equations that don't have real solutions.
The imaginary unit $i$ is necessary when we take the square root of a negative number.
When solving quadratic equations, if the discriminant $(b^2 - 4ac)$ is negative, the solutions will involve $i$.

In our example, the discriminant was $-96$, leading to the square root of a negative number, thus giving us complex solutions.
The final complex solutions were:
$x = \frac{4 + 2\sqrt{6}i}{5}$ and $x = \frac{4 - 2\sqrt{6}i}{5}$.

solving equations

When solving equations, especially quadratics, it's essential to follow a systematic method to ensure all solutions are found.
Let's review the steps:

First, expand both sides of the equation. This means multiplying out any polynomials.
Next, set the equation to zero. Bring all terms to one side, so the equation is in the form $ax^2 + bx + c = 0$.
Then, identify the coefficients a, b, and c. These coefficients will be used in the quadratic formula.
Apply the quadratic formula. Substitute the coefficients into the formula and simplify.
Lastly, simplify the solutions. This might involve working with complex numbers if the discriminant is negative.

Simplifying solutions involves reducing fractions and ensuring the solutions are in their simplest form. Remember, practice makes perfect, so keep practicing these steps with different problems to become more comfortable with the process.

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91影视

Use the quadratic formula to solve each equation. (All solutions for these equations are nonreal complex numbers.) $$ (6 x+1)(x-2)=(x+2)(x-5) $$

Short Answer

Step by step solution

- Expand both sides of the equation

- Set the equation to zero

- Identify coefficients for the quadratic formula

- Apply the quadratic formula

- Write the solution in simplest form

Key Concepts

quadratic formula

complex numbers

solving equations

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