To check the solutions of the quadratic equation \((2x - 7)(x + 9) = 0\) using the table structure you provided, let's substitute the values into the equation and evaluate each step to see if we get a true statement.
Given Values to Check
-
Value = 0
- Substitute: \((2(0) - 7)((0 + 9)) = 0\)
- Step 1: \((-7)(9) = 0\)
- Step 2: \(-63 = 0\)
- True or False Statement? False
-
Value = 3
- Substitute: \((2(3) - 7)((3 + 9)) = 0\)
- Step 1: \((6 - 7)(12) = 0\)
- Step 2: \((-1)(12) = 0\)
- True or False Statement? False
-
Value = -9
- Substitute: \((2(-9) - 7)((-9 + 9)) = 0\)
- Step 1: \((-18 - 7)(0) = 0\)
- Step 2: \((-25)(0) = 0\)
- True or False Statement? True (since any number multiplied by 0 is 0)
-
Value = 9
- Substitute: \((2(9) - 7)((9 + 9)) = 0\)
- Step 1: \((18 - 7)(18) = 0\)
- Step 2: \((11)(18) = 0\)
- True or False Statement? False
Summary Table
| Value | Substituted | Evaluate: Step 1 | Step 2 | True or False Statement? | |-------|----------------------------------------|-----------------------|------------|--------------------------| | 0 | \((2(0)−7)((0+9))\) | \((-7)(9) = 0\) | \(-63 = 0\) | False | | 3 | \((2(3)−7)((3+9))\) | \((-1)(12) = 0\) | \(-12 = 0\) | False | | -9 | \((2(-9)−7)((-9+9))\) | \((-25)(0) = 0\) | \(-0 = 0\) | True | | 9 | \((2(9)−7)((9+9))\) | \((11)(18) = 0\) | \(198 = 0\) | False |
Final Comments
- The only value that satisfies the equation \((2x - 7)(x + 9) = 0\) is \(x = -9\). The solutions to the original quadratic equation can also be found by setting each factor to zero:
- \(2x - 7 = 0 \Rightarrow x = \frac{7}{2}\)
- \(x + 9 = 0 \Rightarrow x = -9\)
So the complete solution set is \(\left{-9, \frac{7}{2}\right}\).
1 answer