Algebra: Concepts & Connections A (MS)

Question

UNIT S Quadratic Equations

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LESSON 10 Quadratic Equations Apply

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OUTLINE

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Place the steps in the correct order to solve the following equation using the quadratic formula.

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(6 points)

Item 12

a = 1, b = - 10, c = - 12

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Z = (((10.2sqrt(37))/2)!)!

x = (- (- 10) plus/minus sqrt((- 10) ^ 2 - 4(1)(- 11)))/(2(1))

x = 5 plus/minus sqrt(37)

\mathcal{Z} = (16k * sqrt(14i))/2

x ^ 2 - 10x = 12

x ^ 2 - 10x - 12 = 0

Answer 1

To solve the quadratic equation \(x^2 - 10x - 12 = 0\) using the quadratic formula, we can outline the steps in the following order:

1. Identify coefficients:

   • Given the equation \(x^2 - 10x - 12 = 0\), we can identify \(a = 1\), \(b = -10\), and \(c = -12\).

2. Write down the quadratic formula:

   • The quadratic formula is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. Substitute the values of \(a\), \(b\), and \(c\):

   • Plugging in values, we get: \[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-12)}}{2(1)} \] Simplifying inside the formula: \[ = \frac{10 \pm \sqrt{100 + 48}}{2} \] \[ = \frac{10 \pm \sqrt{148}}{2} \]

4. Simplify the square root:

   • Since \(\sqrt{148} = \sqrt{4 \times 37} = 2\sqrt{37}\): \[ x = \frac{10 \pm 2\sqrt{37}}{2} \]

5. Divide each term by 2:

   • This simplifies to: \[ x = 5 \pm \sqrt{37} \]

6. Final solutions:

   • The final solutions are: \[ x = 5 + \sqrt{37} \quad \text{and} \quad x = 5 - \sqrt{37} \]

Thus, the correct order of the steps is:

1. Identify coefficients \(a\), \(b\), and \(c\).
2. Write down the quadratic formula.
3. Substitute the values of \(a\), \(b\), and \(c\) into the formula.
4. Simplify and compute inside the square root.
5. Simplify the expression.
6. State the final solutions.