Unit 5, Lesson 19
Deriving the Quadratic Formula
Integrated Math 2
Practice
Problem 1
- The quadratic equation \(x^2 + 7x + 10 = 0\) is in the form of \(ax^2 + bx + c = 0\). What are the values of \(a\), \(b\), and \(c\)?
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Some steps for solving the equation by completing the square have been started here. In the third line, what might be a good reason for multiplying each side of the equation by 4?
\(\displaystyle \begin {align}\\ x^2 + 7x + 10 &= 0 &\hspace{0.1in}& \text {original equation}\\\\ x^2 + 7x &= \text-10 &\hspace{0.1in}& \text {Subtract 10 from each side.}\\\\ 4x^2 + 4(7x) &= 4(\text-10) &\hspace{0.1in}& \text {Multiply each side by 4.}\\\\ (2x)^2 + 2(7)2x + \underline{\hspace{0.3in}}^2 &= \underline{\hspace{0.3in}}^2 - 4(10) &\hspace{0.1in}& \text {Rewrite } 4x^2 \text{ as } (2x)^2\\ &\text{} &\hspace{0.1in}& \text{and }4(7x) \text{ as } 2(7)2x.\\\\ (2x+\underline{\hspace{0.3in}})^2 &= \underline{\hspace{0.3in}}^2 - 4(10)\\\\ 2x+\underline{\hspace{0.3in}} &= \pm \sqrt { \underline{\hspace{0.3in}}^2 - 4(10)}\\\\ 2x &= \underline{\hspace{0.3in}} \pm \sqrt { \underline{\hspace{0.3in}}^2 - 4(10)}\\\\ x &=\\ \end {align}\)
- Complete the unfinished steps, and explain what happens in these steps.
- Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula, \(\displaystyle x = {\text-b \pm \sqrt{b^2-4ac} \over 2a}\), but do not evaluate any of the expressions. Explain how the expression on the right side of the equal sign is related to solving \(x^2+7x+10=0\) by completing the square.