Question

Solve each quadratic equation by completing the square. Give exact answers--no decimals.

Answer 1

Answer:

x= 3/2 +(√47) i / 2  , x = 3/2-(√47) i / 2

Step-by-step explanation:

We have given the equation :

x² -3x +14 = 0

We have to find the value of x.

x² -3x +14 = 0

x²-3x = -14

x² -2(x)(3/2) +(3/2)² = -14 +(3/2)²

(x - 3/2)² = -14 + 9/4

(x - 3/2)²  = -47/4

Squaring both sides we get,

(x-3/2)  = ±√ -47/4

(x-3/2)  = ±(√47) i / 2

x = 3/2 ± (√47) i / 2

x= 3/2 +(√47) i / 2  or x = 3/2-(√47) i / 2 is the answer.

Answer 2

Answer:

$x_1 = \frac{3}{2} + \frac{1}{2}(\sqrt{47})i\\\\x_2 = \frac{3}{2} - \frac{1}{2}(\sqrt{47})i$

Step-by-step explanation:

In this problem we have the equation of the following quadratic equation and we want to solve it using the method of square completion:

$x ^ 2 -3x +14 = 0$

The steps are shown below:

For any equation of the form: $ax ^ 2 + bx + c = 0$

1. If the coefficient a is different from 1, then take a as a common factor.

In this case $a = 1$.

Then we go directly to step 2

2. Take the coefficient b that accompanies the variable x. In this case the coefficient is -3. Then, divide by 2 and the result squared it.

We have:

$\frac{-3}{2} = -\frac{3}{2}\\\\(-\frac{3}{2}) ^ 2 = (\frac{9}{4})$

3. Add the term obtained in the previous step on both sides of equality:

$x ^ 2 -3x + (\frac{9}{4}) = -14 + (\frac{9}{4})$

4. Factor the resulting expression, and you will get:

$(x -\frac{3}{2}) ^ 2 = -(\frac{47}{4})$

Now solve the equation:

Note that the term $(x -\frac{3}{2}) ^ 2$ is always > 0 therefore it can not be equal to $-(\frac{47}{4})$

The equation has no solution in real numbers.

In the same way we can find the complex roots:

$(x -\frac{3}{2}) ^ 2 = -(\frac{47}{4})\\\\x -\frac{3}{2} = \±\sqrt{-(\frac{47}{4})}\\\\x = \frac{3}{2} \±\frac{1}{2}\sqrt{-47}\\\\x = \frac{3}{2} \±\frac{1}{2}(\sqrt{47})i\\\\x_1 = \frac{3}{2} + \frac{1}{2}(\sqrt{47})i\\\\x_2 = \frac{3}{2} - \frac{1}{2}(\sqrt{47})i$