Question

question 1 (5 points)
solve the quadratic equation (4x^2 - x = 8) using the quadratic formula.
a) (x = \frac{1 pm sqrt{129}}{4})
b) (x = \frac{-1 pm sqrt{129}}{8})
c) (x = \frac{-40 pm sqrt{129}}{8})
d) (x = \frac{1 pm sqrt{129}}{8})

question 2 (5 points)
solve the quadratic equation (x^2 - 10x + 14 = -4) by completing the square.
a) (x = 5 pm sqrt{7})
b) (x = -5 pm sqrt{7})
c) (x = 7 pm sqrt{5})
d) (x = -7 pm sqrt{5})

Explanation:

Write in standard form

To solve the quadratic equation \(4x^2 - x = 8\) using the quadratic formula, we first rewrite it in standard form \(ax^2 + bx + c = 0\) by subtracting \(8\) from both sides:

$$4x^2 - x - 8 = 0$$

Here, the coefficients are:

$$a = 4, \quad b = -1, \quad c = -8$$

Apply the quadratic formula

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we substitute our coefficients:

$$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(-8)}}{2(4)}$$

Simplify the terms inside and outside the radical:

$$x = \frac{1 \pm \sqrt{1 - (-128)}}{8}$$

$$x = \frac{1 \pm \sqrt{1 + 128}}{8}$$

$$x = \frac{1 \pm \sqrt{129}}{8}$$

This matches option D for Question 1.

Prepare for completing the square

For Question 2, we need to solve \(x^2 - 10x + 14 = -4\) by completing the square. First, we isolate the variable terms by subtracting \(14\) from both sides:

$$x^2 - 10x = -18$$

Complete the square

To complete the square, we take half of the coefficient of \(x\), square it, and add it to both sides.
Half of \(-10\) is \(-5\), and \((-5)^2 = 25\).

$$x^2 - 10x + 25 = -18 + 25$$

Write the left side as a perfect square and simplify the right side:

$$(x - 5)^2 = 7$$

Solve for x

Take the square root of both sides:

$$x - 5 = \pm\sqrt{7}$$

Add \(5\) to both sides to isolate \(x\):

$$x = 5 \pm \sqrt{7}$$

This matches option A for Question 2.

Answer:

Question 1

• A) \(x = \frac{1 \pm \sqrt{129}}{4}\)
• B) \(x = \frac{-1 \pm \sqrt{129}}{8}\)
• C) \(x = \frac{-40 \pm \sqrt{129}}{8}\)
• D) \(x = \frac{1 \pm \sqrt{129}}{8}\) (Correct answer)

Question 2

• A) \(x = 5 \pm \sqrt{7}\) (Correct answer)
• B) \(x = -5 \pm \sqrt{7}\)
• C) \(x = 7 \pm \sqrt{5}\)
• D) \(x = -7 \pm \sqrt{5}\)