[size=150]A. [math](x+4)(x-3)[/math][br][br]B.[math]3x^2-8x+5[/math][br][br]C. [math]x^2-25[/math][br][br]D. [math]x^2+2x+3[/math][br][br][/size] Explain your reasoning.
[size=150]Here are three quadratic equations, each with two solutions. Find both solutions to each equation, using the zero product property somewhere along the way. Show each step in your reasoning.[/size][br][br][math]x^2=6x[/math]
[math]x\left(x+4\right)=x+4[/math]
[math]2x\left(x-1\right)+3x-3=0[/math]
[size=150]An engineer is designing a fountain that shoots out drops of water. The nozzle from which the water is launched is 3 meters above the ground. It shoots out a drop of water at a vertical velocity of 9 meters per second.[br][br]Function [math]h[/math] models the height in meters, [math]h[/math], of a drop of water [math]t[/math] seconds after it is shot out from the nozzle. The function is defined by the equation [math]h\left(t\right)=-5t^2+9t+3[/math].[/size][br][br][size=150]How many seconds until the drop of water hits the ground?[/size][br]Write an equation that we could solve to answer the question.
Try to solve the equation by writing the expression in factored form and using the zero product property.[br]
Explain how you found the solution.[br]
[math]9x^2+21x+10 \\ [br](3x)^2+7(3x)+10 \\ [br]N^2+7N+10\\ [br](N+2)(N+5) \\[br]3x+2)(3x+5)[/math][br][br]Use the distributive property to expand [math]\left(3x+2\right)\left(3x+5\right)[/math]. Show your reasoning and write the resulting expression in standard form. Is it equivalent to [math]9x^2+21x+10[/math]?
Study the method and make sense of what was done in each step. Make a note of your thinking and be prepared to explain it.[br]
Try the method to write each of these expressions in factored form:[br][br][list][*][math]4x^2+28x+45[/math][/*][/list]
[list][*][math]25x^2-35x+6[/math][/*][/list]
[size=100]You have probably noticed that the coefficient of the squared term in all of the previous examples is a perfect square. [/size]What if that coefficient is not a perfect square?
Here is an example of an expression whose squared term has a coefficient that is not a squared term.[br][br][math]5x^2+17x+6 \\ [br]\frac{1}{5} \cdot 5 \cdot (5x^2 + 17x + 6)\\ [br]\frac{1}{5} (25x^2 + 85x + 30) \\[br]\frac{1}{5} ((5x)^2 + 17 (5x) + 30)\\[br]\frac{1}{5} (N^2 + 17N + 30)\\[br]\frac{1}{5} (N+15)(N+2) \\ [br]\frac{1}{5} (5x+15)(5x+2) \\[br](x+3)(5x+2)[/math][br][br]Use the distributive property to expand [math]\left(x+3\right)\left(5x+2\right)[/math]. Show your reasoning and write the resulting expression in standard form. Is it equivalent to [math]5x^2+17x+6[/math]?
Study the method and make sense of what was done in each step and why. Make a note of your thinking and be prepared to explain it.[br]
Try the method to write each of these expressions in factored form:[br][list][*][math]3x^2+16x+5[/math][/*][/list]
[list][*][math]10x^2-41x+4[/math][/*][/list]