What are the Solutions?
1) In each of the following equations, the variables represent real numbers. Assuming each equation is true, what can you conclude about the values of the variables? Explain each step in your reasoning.
a. \(2z+3=0\)
b. \(7x=0\)
c. \(7(y-5)=0\)
d. \(ab=0\)
2) The Zero Product Property states that if the product of two numbers is zero then at least one of the numbers is zero. In symbols, if \(ab=0\), then \(a=0\) or \(b=0\). We can use this property when we solve equations where a product is zero. Use the Zero Product Property to find all solutions. Explain each step in your reasoning.
a. \(x(13-4x)=0\)
b. \(7(y+12)=0\)
c. \((x-19)(x+13)=0\)
d. \((y-6)(3z-4)=0\)
3) Explain how the property can be used to find both solutions to each of the following equations, and explain each step in your reasoning.
a. \((x-1)(x-3)=0\)
b. \(2x(x-1)+3x-3=0\)
c. \(x+4=x(x+4)\)
d. \(x^2=6x\)
e. \(x^2+10=7x\)
Solutions
Problems adapted from illustrativemathematics.org.
1) In each of the following equations, the variables represent real numbers. Assuming each equation is true, what can you conclude about the values of the variables? Explain each step in your reasoning.
a. \(2z+3=0\)
b. \(7x=0\)
c. \(7(y-5)=0\)
d. \(ab=0\)
2) The Zero Product Property states that if the product of two numbers is zero then at least one of the numbers is zero. In symbols, if \(ab=0\), then \(a=0\) or \(b=0\). We can use this property when we solve equations where a product is zero. Use the Zero Product Property to find all solutions. Explain each step in your reasoning.
a. \(x(13-4x)=0\)
b. \(7(y+12)=0\)
c. \((x-19)(x+13)=0\)
d. \((y-6)(3z-4)=0\)
3) Explain how the property can be used to find both solutions to each of the following equations, and explain each step in your reasoning.
a. \((x-1)(x-3)=0\)
b. \(2x(x-1)+3x-3=0\)
c. \(x+4=x(x+4)\)
d. \(x^2=6x\)
e. \(x^2+10=7x\)
Solutions
Problems adapted from illustrativemathematics.org.