To solve the quadratic equation [tex]\( x^2 - 4x + 3 = 0 \)[/tex] using the quadratic formula, we follow these steps:
1. Identify the coefficients:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here,
[tex]\( a = 1 \)[/tex],
[tex]\( b = -4 \)[/tex], and
[tex]\( c = 3 \)[/tex].
2. Write down the quadratic formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
The discriminant, [tex]\( \Delta \)[/tex], is given by [tex]\( b^2 - 4ac \)[/tex].
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (-4)^2 - 4 \cdot 1 \cdot 3 = 16 - 12 = 4
\][/tex]
4. Determine the roots using the quadratic formula:
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and the discriminant [tex]\( \Delta \)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{-(-4) \pm \sqrt{4}}{2 \cdot 1} = \frac{4 \pm 2}{2}
\][/tex]
5. Evaluate the expressions for both roots:
- For the addition part ([tex]\( \frac{4 + 2}{2} \)[/tex]):
[tex]\[
x_1 = \frac{4 + 2}{2} = \frac{6}{2} = 3
\][/tex]
- For the subtraction part ([tex]\( \frac{4 - 2}{2} \)[/tex]):
[tex]\[
x_2 = \frac{4 - 2}{2} = \frac{2}{2} = 1
\][/tex]
6. Describe the solution:
The discriminant is 4, which is a positive number, indicating that there are two distinct real roots. Therefore, the quadratic equation [tex]\( x^2 - 4x + 3 = 0 \)[/tex] has the following solutions:
[tex]\[
x_1 = 3 \quad \text{and} \quad x_2 = 1
\][/tex]
To summarize, the roots of the quadratic equation [tex]\( x^2 - 4x + 3 = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = 1 \)[/tex].
