Question

quadratic equations by factoring
period __ date ___

1. ((4x - 1)(5x + 10)=0)
2. (3x(9x - 3)=0)
3. ((2x + 4)^2 = 0)
4. (7(x + 3)^2 = 0)

Explanation:

To solve quadratic equations by factoring, we use the Zero - Product Property, which states that if \(ab = 0\), then either \(a=0\) or \(b = 0\) (or both). We will solve each equation one by one.

Equation 2: \((4x - 1)(5x+10)=0\)

Step 1: Apply the Zero - Product Property

Set each factor equal to zero:
\(4x - 1=0\) or \(5x + 10=0\)

Step 2: Solve \(4x - 1=0\) for \(x\)

Add 1 to both sides of the equation:
\(4x-1 + 1=0 + 1\)
\(4x=1\)
Divide both sides by 4:
\(x=\frac{1}{4}\)

Step 3: Solve \(5x + 10=0\) for \(x\)

Subtract 10 from both sides:
\(5x+10 - 10=0 - 10\)
\(5x=- 10\)
Divide both sides by 5:
\(x=-2\)

Equation 3: \(3x(9x - 3)=0\)

Step 1: Apply the Zero - Product Property

Set each factor equal to zero:
\(3x = 0\) or \(9x-3=0\)

Step 2: Solve \(3x = 0\) for \(x\)

Divide both sides by 3:
\(x = 0\)

Step 3: Solve \(9x-3=0\) for \(x\)

Add 3 to both sides:
\(9x-3 + 3=0 + 3\)
\(9x=3\)
Divide both sides by 9:
\(x=\frac{3}{9}=\frac{1}{3}\)

Equation 5: \((2x + 4)^{2}=0\)

Step 1: Rewrite the equation

We can write \((2x + 4)^{2}\) as \((2x + 4)(2x + 4)=0\)

Step 2: Apply the Zero - Product Property

Set \(2x + 4=0\) (since both factors are the same)
Subtract 4 from both sides:
\(2x+4 - 4=0 - 4\)
\(2x=-4\)
Divide both sides by 2:
\(x=-2\)

Equation 6: \(7(x + 3)^{2}=0\)

Step 1: Divide both sides by 7

\(\frac{7(x + 3)^{2}}{7}=\frac{0}{7}\)
\((x + 3)^{2}=0\)

Step 2: Rewrite the equation

We can write \((x + 3)^{2}\) as \((x + 3)(x + 3)=0\)

Step 3: Apply the Zero - Product Property

Set \(x + 3=0\) (since both factors are the same)
Subtract 3 from both sides:
\(x+3 - 3=0 - 3\)
\(x=-3\)

Final Answers:

• For equation \((4x - 1)(5x + 10)=0\): \(x=\frac{1}{4}\) or \(x=-2\)
• For equation \(3x(9x - 3)=0\): \(x = 0\) or \(x=\frac{1}{3}\)
• For equation \((2x + 4)^{2}=0\): \(x=-2\)
• For equation \(7(x + 3)^{2}=0\): \(x=-3\)

Answer:

To solve quadratic equations by factoring, we use the Zero - Product Property, which states that if \(ab = 0\), then either \(a=0\) or \(b = 0\) (or both). We will solve each equation one by one.

Equation 2: \((4x - 1)(5x+10)=0\)

Step 1: Apply the Zero - Product Property

Set each factor equal to zero:
\(4x - 1=0\) or \(5x + 10=0\)

Step 2: Solve \(4x - 1=0\) for \(x\)

Add 1 to both sides of the equation:
\(4x-1 + 1=0 + 1\)
\(4x=1\)
Divide both sides by 4:
\(x=\frac{1}{4}\)

Step 3: Solve \(5x + 10=0\) for \(x\)

Subtract 10 from both sides:
\(5x+10 - 10=0 - 10\)
\(5x=- 10\)
Divide both sides by 5:
\(x=-2\)

Equation 3: \(3x(9x - 3)=0\)

Step 1: Apply the Zero - Product Property

Set each factor equal to zero:
\(3x = 0\) or \(9x-3=0\)

Step 2: Solve \(3x = 0\) for \(x\)

Divide both sides by 3:
\(x = 0\)

Step 3: Solve \(9x-3=0\) for \(x\)

Add 3 to both sides:
\(9x-3 + 3=0 + 3\)
\(9x=3\)
Divide both sides by 9:
\(x=\frac{3}{9}=\frac{1}{3}\)

Equation 5: \((2x + 4)^{2}=0\)

Step 1: Rewrite the equation

We can write \((2x + 4)^{2}\) as \((2x + 4)(2x + 4)=0\)

Step 2: Apply the Zero - Product Property

Set \(2x + 4=0\) (since both factors are the same)
Subtract 4 from both sides:
\(2x+4 - 4=0 - 4\)
\(2x=-4\)
Divide both sides by 2:
\(x=-2\)

Equation 6: \(7(x + 3)^{2}=0\)

Step 1: Divide both sides by 7

\(\frac{7(x + 3)^{2}}{7}=\frac{0}{7}\)
\((x + 3)^{2}=0\)

Step 2: Rewrite the equation

We can write \((x + 3)^{2}\) as \((x + 3)(x + 3)=0\)

Step 3: Apply the Zero - Product Property

Set \(x + 3=0\) (since both factors are the same)
Subtract 3 from both sides:
\(x+3 - 3=0 - 3\)
\(x=-3\)

Final Answers:

• For equation \((4x - 1)(5x + 10)=0\): \(x=\frac{1}{4}\) or \(x=-2\)
• For equation \(3x(9x - 3)=0\): \(x = 0\) or \(x=\frac{1}{3}\)
• For equation \((2x + 4)^{2}=0\): \(x=-2\)
• For equation \(7(x + 3)^{2}=0\): \(x=-3\)