Unit 5, Lesson 3
Solving Quadratic Equations by Reasoning
Integrated Math 2
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Consider the equation \(x^2=9\).
Solve \((x-1)^2 = 16\). Explain or show your reasoning.
Here is one way to solve the equation \(\frac59 y^2 = 5\). Explain what is done in each step.
\(\begin{align}\frac59y^2 &= 5 &\quad &\text{Original equation}\\5y^2&=45 &\quad &\text{Step 1} \\\\y^2&=9 &\quad& \text{Step 2} \\\\y=3 \qquad &\text{or} \qquad y=\text-3 &\quad& \text{Step 3} \end{align}\)
Diego and Jada are working together to solve the quadratic equation \((x-2)^2 = 100\).
Diego solves the equation by dividing each side of the equation by 2 and then adding 2 to each side. He writes:
\(\displaystyle \begin{align} (x-2)&=50\\ x&=52\\ \end{align}\)
Jada asks Diego why he divides each side by 2 and he says, “I want to find a number that equals 100 when multiplied by itself. That number is half of 100.”
A billboard installer accidentally drops a tool while working on a billboard. The height of the tool \(t\) seconds after it is dropped is given by the function \(h(t) = 115-16t^2\), where \(h\) is in feet.
A zoo offers unlimited drink refills to visitors who purchase its souvenir cup. The cup and the first fill cost \$10, and refills after that are \$2 each. The expression \(10+2r\) represents the total cost of the cup and \(r\) refills.