To solve the quadratic equation [tex]\( 4x^2 - 20x + 21 = 0 \)[/tex] by factoring, we need to follow these steps:
### Step 1: Write the Quadratic Equation in Standard Form
The given quadratic equation is already in the standard form:
[tex]\[ 4x^2 - 20x + 21 = 0 \][/tex]
### Step 2: Factor the Quadratic Expression
We need to express the quadratic expression [tex]\( 4x^2 - 20x + 21 \)[/tex] as a product of two binomials.
By factoring, we get:
[tex]\[ 4x^2 - 20x + 21 = (2x - 7)(2x - 3) \][/tex]
### Step 3: Set Each Factor Equal to Zero
Now that we have the quadratic equation factored, we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 7 = 0 \quad \text{or} \quad 2x - 3 = 0 \][/tex]
### Step 4: Solve Each Equation
Solve the first equation:
[tex]\[ 2x - 7 = 0 \][/tex]
Add 7 to both sides:
[tex]\[ 2x = 7 \][/tex]
Divide both sides by 2:
[tex]\[ x = \frac{7}{2} \][/tex]
Solve the second equation:
[tex]\[ 2x - 3 = 0 \][/tex]
Add 3 to both sides:
[tex]\[ 2x = 3 \][/tex]
Divide both sides by 2:
[tex]\[ x = \frac{3}{2} \][/tex]
### Conclusion
The solutions to the quadratic equation [tex]\( 4x^2 - 20x + 21 = 0 \)[/tex] are:
[tex]\[ x = \frac{3}{2} \quad \text{and} \quad x = \frac{7}{2} \][/tex]
Thus, the correct answer is:
a. [tex]\( x=3/2 \)[/tex] and [tex]\( x=7/2 \)[/tex]
