Problem 31 Solve by completing the square. ... [FREE SOLUTION]

Chapter 11: Problem 31

Solve by completing the square. $$x^{2}-7 x+12=0$$

Short Answer

Expert verified

The solutions to the given quadratic equation $x^2 - 7x + 12 = 0$ using the completing the square method are: $x = \frac{7}{2} + \frac{i}{2}$ and $x = \frac{7}{2} - \frac{i}{2}$.

Step by step solution

Write the equation in the form of (x-a)^2 = b

Let's rewrite the given quadratic equation and try to form a perfect square trinomial: $x^2 - 7x + 12 = 0$ To rewrite the equation in the form $(x-a)^2 = b$, we need to add and subtract the square of half of the coefficient of the x term: Half of the coefficient of x term = $\frac{-7}{2}$ Square of half of the coefficient of x term = $\left(\frac{-7}{2}\right)^2 = \frac{49}{4}$ Now, subtract $49/4$ and add $49/4$: $x^2 - 7x + \frac{49}{4} - \frac{49}{4} + 12 = 0$ Add the terms in the trinomial: $\left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + 12 = 0$ Now, we need to find the value of $b$:

Find the value of b

Calculate the value of b: $-\frac{49}{4} + 12 = -\frac{49}{4} + \frac{48}{4} = -\frac{1}{4}$ The equation becomes: $\left(x - \frac{7}{2}\right)^2 = -\frac{1}{4}$

Solve the equation for x

Now we have the equation in the form $(x-a)^2 = b$: $\left(x - \frac{7}{2}\right)^2 = -\frac{1}{4}$ Take the square root of both sides: $x - \frac{7}{2} = \pm \sqrt{-\frac{1}{4}}$ We can write the negative term inside the square root as $i$, where $i^2 = -1$: $x - \frac{7}{2} = \pm \frac{i}{2}$ Now, add $\frac{7}{2}$ to both sides to solve for x: $x = \frac{7}{2} \pm \frac{i}{2}$ So, the two solutions to the given quadratic equation are: $x = \frac{7}{2} + \frac{i}{2}$ and $x = \frac{7}{2} - \frac{i}{2}$

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Key Concepts

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

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form \[ ax^2 + bx + c = 0 \]where $a$, $b$, and $c$ are constants and $a eq 0$.

The equation involves the variable $x^2$ as the highest degree term.

It is "quadratic" because "quad" is derived from "quadrus," meaning "square."

For the given equation $x^2 - 7x + 12 = 0$, we recognize it as a standard quadratic form where $a = 1$, $b = -7$, and $c = 12$. This form allows for various techniques to find its roots, or solutions. One effective method is "completing the square," which transforms the quadratic equation into a perfect square trinomial.

Imaginary Numbers

Imaginary numbers are used when dealing with square roots of negative numbers. Normally, squaring a real number cannot result in a negative value. However, certain quadratic equations have negative terms under a square root, which leads us to imaginary numbers. They are introduced using the imaginary unit $ i $, where $ i^2 = -1 $.

Imaginary numbers are in the form of $bi$ where $b$ is a real number.

This concept helps to extend the real numbers to complex numbers, allowing for meaningful solutions in previously unsolvable scenarios.

In our solution from the exercise, when we reached the equation \[ \left(x - \frac{7}{2}\right)^2 = -\frac{1}{4} \]we saw the need for this concept.
The negative sign under the square root gives rise to the imaginary unit $i$, resulting in terms like $ \frac{i}{2} $. This transformation enables us to handle negative square roots effectively.

Complex Solutions

Complex solutions refer to solutions that involve both real and imaginary numbers. A complex number is generally expressed in the form $a + bi$,where $a$ is the real part and $bi$ is the imaginary part.

In the quadratic equation context, complex solutions occur when the discriminant (from the quadratic formula) is negative.

This forces the solution into the realm of imaginary numbers, producing complex values as roots.

For the quadratic equation in the exercise, the final solutions were $x = \frac{7}{2} + \frac{i}{2}$and $x = \frac{7}{2} - \frac{i}{2}$.
Here, $\frac{7}{2}$ is the real part and $\pm \frac{i}{2}$ is the imaginary part.
Complex solutions like these exemplify the breadth of solving quadratic equations, indicating that not all solutions are purely real numbers.

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91影视

Solve by completing the square. $$x^{2}-7 x+12=0$$

Short Answer

Step by step solution

Write the equation in the form of (x-a)^2 = b

Find the value of b

Solve the equation for x

Key Concepts

Quadratic Equation

Imaginary Numbers

Complex Solutions

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