What are the solutions to the equation \((10-x)(3x-9)=0\)?

Solve each equation.

1. \((x-6)(x+5)=0\)
2. \((x-3)(\frac23 x - 6)=0\)
3. \((\text-3x-15)(x+7)=0\)

Consider the expressions \((x-4)(3x-6)\) and \(3x^2 - 18x + 24\).

Show that the two expressions define the same function.

Kiran saw that if the equation \((x+2)(x-4)=0\) is true, then by the zero product property, either \(x+2\) is 0 or \(x-4\) is 0. He then reasoned that, if \((x+2)(x-4)=72\) is true, then either \(x+2\) is equal to 72 or \(x-4\) is equal to 72. 

Explain why Kiran’s conclusion is incorrect.

Andre wants to solve the equation \(5x^2-4x-18=20\). He uses a graphing calculator to graph \(y=5x^2-4x-18\) and \(y=20\) and finds that the graphs cross at the points \((\text-2.39, 20)\) and \((3.19, 20)\).

1. Substitute each \(x\)-value that Andre found into the expression \(5x^2-4x-18\). Then evaluate the expression.
2. Why did neither solution make \(5x^2-4x-18\) equal exactly 20?

Select all the solutions to the equation \(7x^2 = 343\).

Han says this pattern of dots can be represented by a quadratic relationship because the dots are arranged in a square in each step.

Do you agree? Explain your reasoning.